How Do Marbles Play In Removing Illiteracy
TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
Example 2
405
Self Tutor
John plays Peter at tennis. The first to win two sets wins the match.
Illustrate the sample space using a tree diagram.
If J means “John wins the set” and P means “Peter wins the set” then the tree diagram is:
1st set
2nd set
3rd set
J
J
J
P
P
J
We could write the sample space in set notation as
S = fJJ, JPJ, JPP, PJJ, PJP, PPg.
J
P
P
P
2 Use a tree diagram to illustrate the sample space for the following: a The genders of a 4-child family. b Bag A contains red and white marbles and bag B contains blue and yellow marbles.
A bag is selected and one marble is taken from it. c Hats A, B and C each contain pink and purple tickets. A hat is selected and …show more content…
then two tickets are taken from it. d Two teams, X and Y, play football. The first team to kick 3 goals wins the match. e Jody and Petria play tennis. The first to win two sets in a row or a total of 3 sets, wins the match.
3 A bag contains five marbles. One is blue, one is red, and three are green. Two marbles are selected from the bag without replacement. Use a tree diagram to show all possible outcomes. B PROBABILITIES FROM TREE DIAGRAMS
Tree diagrams can be used to illustrate sample spaces providing there are not too many different alternatives. Once the sample space has been illustrated, the tree diagram can be used to determine probabilities.
INDEPENDENT EVENTS of passing Physics and
We assume that these events are independent, which means the probability of Charles passing
Physics is not related in any way to him passing Chemistry.
magenta
P
Qt_
F
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A tree diagram showing all possible outcomes is shown alongside with P being a pass and F being a fail.
cyan
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Physics
50
4
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75
Suppose Charles has probability
black
of passing Chemistry.
Chemistry
outcomes probabilities
Jq_p_
P
PP
Rt_ ´ Jq_p_ = Wt_Ip_
Dq_p_
F
PF
Rt_ ´ Dq_p_ = Qt_Wp_
Jq_p_
P
FP
Qt_ ´ Jq_p_ = Jt_p_
Dq_p_
F
FF
Qt_ ´ Dq_p_ = Dt_p_
IB MYP_4
406
TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
The probabilities for passing and failing each subject are marked along the branches.
When using tree diagrams to solve probability questions:
² the probability for each branch is calculated by multiplying the probabilities along that branch ² if two or more branches meet the description of the compound event, the probability of each branch is found and then they are added.
Example 3
Self Tutor
Jessica has probability 45 of getting an A in Mathematics and probability an A for English. a Display this information on a tree diagram. b What is the probability Jessica gets one A in the two subjects?
2
5
of getting
a A is the event of getting an A. N is the event of not getting an A.
English
outcomes
Wt_
A
AA
Et_
N
AN
Rt_ ´ Et_ = Qw_Wt_
Wt_
A
NA
Qt_ ´ Wt_ = Sw_t_
Et_
N
NN
Mathematics
Rt_
A
Qt_
N
probabilities
b Jessica can get one A either by getting an A in Mathematics but not in English, or by getting an A in English but not in Mathematics.
So, the total probability is 45 £ 35 + 15 £ 25 = 14
25
These events are highlighted on the tree diagram.
EXERCISE 20B.1
1 A spinner has probability 13 of finishing on blue and probability 23 of finishing on green.
Xin spins the spinner two times. a Display this information on a tree diagram. b Find the probability that the spinner finishes on blue once and on green once.
cyan
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2 Andrew has probability 15 of winning the 100 metre sprint and probability the 200 metre race. a Draw a tree diagram showing all of Andrew’s chances. b What is the probability Andrew wins both races? c What is the probability that Andrew wins exactly one of the two races? d What is the probability Andrew loses both races? e Find the sum of the probabilities in b, c and d.
Explain why this answer is to be expected.
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IB MYP_4
407
TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
DEPENDENT EVENTS
Suppose a hat contains 5 red and 3 blue tickets. One ticket is randomly chosen, its colour is noted, and it is then thrown away. A second ticket is randomly selected. What is the chance that it is red?
P(second is red) =
4
7
4 reds remaining
7 to choose from
If the first ticket was blue, P(second is red) =
5
7
5 reds remaining
7 to choose from
If the first ticket was red,
R
B
B
R
R
R
B
R
So, the event of the second ticket being red depends on what colour the first ticket was.
In such a case we say we have dependent events.
Two or more events are dependent if they are not independent.
Dependent events are events where the occurrence of one of the events does affect the occurrence of the other event.
If A and B are dependent events then
P(A then B) = P(A) £ P(B given that A has occurred).
A typical example of dependent events is when we sample two objects without replacement.
This means that the first object is not replaced before the second is selected. It therefore cannot be selected twice.
Example 4
Self Tutor
A box contains 4 red and 2 yellow tickets. Two tickets are randomly selected one after the other from the box, without replacement. a Display this information on a tree diagram. b What is the probability that both are red? c What is the probability that one is red and the other is yellow? a Let R be the event that a red ticket is drawn and Y be the event that a yellow ticket is drawn.
Note that the outcome of the second event depends on the first.
1st selection
Ry_
R
Wy_
Y
2nd selection
Et_
R
Ry_ ´ Et_
Wt_
Y
Ry_ ´ Wt_
Rt_
R
Wy_ ´ Rt_
Qt_
Y
Wy_ ´ Qt_
b P(R and then R) = P(R) £ P(R given that R has occurred)
=
4
6
£
3
5
=
2
5
P(one is R and the other is Y )
= P(R and then Y or Y and then R)
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8
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IB MYP_4
408
TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
EXERCISE 20B.2
1 A box contains 7 red and 3 green balls. Two balls are randomly selected from the box one after the other. The first is not replaced in the box before the second is selected.
Determine the probability that: a both are red b the first is green and the second is red c a green and a red are obtained.
Selecting items simultaneously means there is no replacement.
2 A hat contains the names of the 7 players in a tennis squad including the captain and the vice captain. A team of 3 players is chosen at random by drawing the names from the hat. a Display this information on a tree diagram.
Distinguish between the captain, the vice captain, and the other players. b Find the probability that the team: i does not contain the captain ii contains neither the captain nor the vice captain iii contains either the captain or the vice captain, but not both.
3 Amelie has a bag containing two different varieties of apples. They are approximately the same size and shape, but one variety is red and the other is green. There are 4 red apples and 6 green ones. She selects one apple at random, eats it, and then takes another, also at random. Determine the probability that: a both apples were red b both apples were green c the first was red and the second was green d the first was green and the second was red e she ate one red and one green apple.
4 Marjut has a carton containing 10 cans of soup. 4 cans are tomato and the rest are pumpkin. She selects 2 cans at random without looking at the labels. a Let T represent tomato and P represent pumpkin. Draw a tree diagram to illustrate this sampling process. b What is the probability that both cans were tomato soup? c What is the probability that one can was tomato and the other can was pumpkin soup?
LARGE SAMPLE SPACES
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Sometimes the number of possible outcomes is sufficiently large that it is a waste of time to draw a tree diagram. We can still use the same principles we used with the tree diagrams to calculate the probabilities.
black
IB MYP_4
TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
Example 5
409
Self Tutor
A hat contains 20 tickets with the numbers 1, 2, 3, ..., 19 and 20 printed on them.
If 3 tickets are drawn from the hat without replacement, determine the probability that all are prime numbers.
There are 20 numbers of which 8 are primes: f2, 3, 5, 7, 11, 13, 17, 19g
) P(3 primes)
= P(1st drawn is prime and 2nd is prime and 3rd is prime)
8
7
6
£ 19
£ 18
= 20
8 primes out of 20 numbers
7 primes out of 19 numbers after a successful first draw
6 primes out of 18 numbers after two successful draws
¼ 0:0491
EXERCISE 20B.3
1 A lottery has 100 tickets which are placed in a barrel. Two tickets are drawn at random from the barrel to decide 2 prizes. If John has 2 tickets in the lottery, determine his probability of winning: a first prize
b first and second prize
c second prize but not first prize.
d none of the prizes.
2 A bin contains 12 identically shaped chocolates of which 8 are strawberry creams. If 3 chocolates are selected at random from the bin, determine the probability that: a they are all strawberry creams
b none of them are strawberry creams.
3 A bag contains two white and five red marbles. Three marbles are selected simultaneously. Determine the probability that: a all are red
b only two are red
c at least two are red.
SELECTION WITH AND WITHOUT REPLACEMENT
You may have noticed that when we work with tree diagrams, the probabilities of independent and dependent events are calculated using the same method. The following example compares two such events.
Consider a box containing 3 red, 2 blue and 1 green marble. Suppose we wish to sample two marbles: ² with replacement of the first before the second is drawn
² without replacement of the first before the second is drawn.
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Examine how the tree diagrams differ:
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IB MYP_4
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TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
With replacement
Without replacement
1st
Ey_
R
Wy_
2nd
R
B ( )
Qy_
Ey_
G
R ( )
Ey_
B
Wy_
Ey_
G
R
Qy_
Wy_
B
Qy_
G
Ey_
Wy_
Wy_
B
Qy_
Qy_
G
This branch represents Blue with the first draw and Red with the second draw. This is written as BR.
Notice that:
Wt_
R
Wt_
B ( )
Qt_
G
Et_
R ( )
Qt_
B
B
Qt_
Et_
Wt_
G
² with replacement
P(two reds) =
² without replacement
P(two reds) =
3
6
3
6
£
£
3
6
2
5
=
=
2nd
R
1st
G
R
G can’t have
GG
1
4
1
5
We can thus see why replacement is important.
Example 6
Self Tutor
For the example above of the box containing 3 red, 2 blue and 1 green marble, find the probability of getting a red and a blue:
a with replacement
b without replacement.
a
b
P(a red and a blue)
= P(RB or BR)
=
=
=
3
2
6 £ 6
12
36
1
3
+
2
6
£
P(a red and a blue)
fticked onesg
3
6
= P(RB or BR)
3
2
6 £ 5
12
30
2
5
=
=
=
+
2
6
£
3
5
fcrossed onesg
EXERCISE 20B.4
Use tree diagrams to help answer the following questions:
1 Jar A contains 4 blue and 2 red marbles. Jar B contains 1 blue and 5 red marbles. A jar is randomly selected and one marble is taken from it. Determine the probability that the marble is blue.
2 Two marbles are drawn in succession from a box containing 2 purple and 5 green marbles.
Determine the probability that the two marbles are different colours if: a the first is replaced
b the first is not replaced.
3 5 tickets numbered 1, 2, 3, 4 and 5 are placed in a bag. Two are taken from the bag without replacement. Determine the probability that:
cyan
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c one is odd and the other is even.
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b both are even
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a both are odd
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IB MYP_4
TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
4 Jar A contains 3 red and 2 green tickets. Jar B contains 3 red and
7 green tickets. A die has 4 faces marked A and 2 faces marked B.
The die is rolled and the result is used to select either jar A or jar B.
When a jar has been selected, two tickets are randomly selected from it without replacement. Determine the probability that: a both are green
411
A
A
B
b they are different in colour.
Example 7
Self Tutor
A bag contains 5 red and 3 blue marbles. Two marbles are drawn simultaneously from the bag. Determine the probability that at least one is red.
P(at least one red) = P(RR or RB or BR)
draw 2 draw 1
Ru_
R
=
Eu_
B
=
Tu_
R
Wu_
B
R
Ti_
Ei_
=
=
B
5
4
5
8 £ 7 + 8
20+15+15
56
50
56
25
28
£
3
7
+
3
8
£
5
7
Alternatively, P(at least one red) = 1 ¡ P(no reds) fcomplementary eventsg
= 1 ¡ P(BB) and so on.
5 A bag contains four red and two blue marbles. Three marbles are selected simultaneously.
Determine the probability that: a all are red
b only two are red
c at least two are red.
6 Box A contains 3 red and 4 green marbles.
Box B contains 5 red and 2 green marbles.
One marble is randomly selected from box A and
A
its colour noted. If it is red, 2 reds are added to box B. If it is green, 2 greens are added to box B.
B
A marble is then selected from box B.
Find the probability that the marble selected from box B is green.
C
BINOMIAL PROBABILITIES
In many situations there are only two possible outcomes. For example:
² a tennis player either wins or loses a game
² you either catch a bus or you don’t
² you either make a free throw or you miss.
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Binomial experiments are concerned with the repetition of several independent trials where there are only two possible outcomes, success and failure. Each trial has the same probability of success.
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IB MYP_4
Example 2
405
Self Tutor
John plays Peter at tennis. The first to win two sets wins the match.
Illustrate the sample space using a tree diagram.
If J means “John wins the set” and P means “Peter wins the set” then the tree diagram is:
1st set
2nd set
3rd set
J
J
J
P
P
J
We could write the sample space in set notation as
S = fJJ, JPJ, JPP, PJJ, PJP, PPg.
J
P
P
P
2 Use a tree diagram to illustrate the sample space for the following: a The genders of a 4-child family. b Bag A contains red and white marbles and bag B contains blue and yellow marbles.
A bag is selected and one marble is taken from it. c Hats A, B and C each contain pink and purple tickets. A hat is selected and …show more content…
then two tickets are taken from it. d Two teams, X and Y, play football. The first team to kick 3 goals wins the match. e Jody and Petria play tennis. The first to win two sets in a row or a total of 3 sets, wins the match.
3 A bag contains five marbles. One is blue, one is red, and three are green. Two marbles are selected from the bag without replacement. Use a tree diagram to show all possible outcomes. B PROBABILITIES FROM TREE DIAGRAMS
Tree diagrams can be used to illustrate sample spaces providing there are not too many different alternatives. Once the sample space has been illustrated, the tree diagram can be used to determine probabilities.
INDEPENDENT EVENTS of passing Physics and
We assume that these events are independent, which means the probability of Charles passing
Physics is not related in any way to him passing Chemistry.
magenta
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Qt_
F
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A tree diagram showing all possible outcomes is shown alongside with P being a pass and F being a fail.
cyan
7
10
Physics
50
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5
75
Suppose Charles has probability
black
of passing Chemistry.
Chemistry
outcomes probabilities
Jq_p_
P
PP
Rt_ ´ Jq_p_ = Wt_Ip_
Dq_p_
F
PF
Rt_ ´ Dq_p_ = Qt_Wp_
Jq_p_
P
FP
Qt_ ´ Jq_p_ = Jt_p_
Dq_p_
F
FF
Qt_ ´ Dq_p_ = Dt_p_
IB MYP_4
406
TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
The probabilities for passing and failing each subject are marked along the branches.
When using tree diagrams to solve probability questions:
² the probability for each branch is calculated by multiplying the probabilities along that branch ² if two or more branches meet the description of the compound event, the probability of each branch is found and then they are added.
Example 3
Self Tutor
Jessica has probability 45 of getting an A in Mathematics and probability an A for English. a Display this information on a tree diagram. b What is the probability Jessica gets one A in the two subjects?
2
5
of getting
a A is the event of getting an A. N is the event of not getting an A.
English
outcomes
Wt_
A
AA
Et_
N
AN
Rt_ ´ Et_ = Qw_Wt_
Wt_
A
NA
Qt_ ´ Wt_ = Sw_t_
Et_
N
NN
Mathematics
Rt_
A
Qt_
N
probabilities
b Jessica can get one A either by getting an A in Mathematics but not in English, or by getting an A in English but not in Mathematics.
So, the total probability is 45 £ 35 + 15 £ 25 = 14
25
These events are highlighted on the tree diagram.
EXERCISE 20B.1
1 A spinner has probability 13 of finishing on blue and probability 23 of finishing on green.
Xin spins the spinner two times. a Display this information on a tree diagram. b Find the probability that the spinner finishes on blue once and on green once.
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2 Andrew has probability 15 of winning the 100 metre sprint and probability the 200 metre race. a Draw a tree diagram showing all of Andrew’s chances. b What is the probability Andrew wins both races? c What is the probability that Andrew wins exactly one of the two races? d What is the probability Andrew loses both races? e Find the sum of the probabilities in b, c and d.
Explain why this answer is to be expected.
black
1
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of winning
IB MYP_4
407
TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
DEPENDENT EVENTS
Suppose a hat contains 5 red and 3 blue tickets. One ticket is randomly chosen, its colour is noted, and it is then thrown away. A second ticket is randomly selected. What is the chance that it is red?
P(second is red) =
4
7
4 reds remaining
7 to choose from
If the first ticket was blue, P(second is red) =
5
7
5 reds remaining
7 to choose from
If the first ticket was red,
R
B
B
R
R
R
B
R
So, the event of the second ticket being red depends on what colour the first ticket was.
In such a case we say we have dependent events.
Two or more events are dependent if they are not independent.
Dependent events are events where the occurrence of one of the events does affect the occurrence of the other event.
If A and B are dependent events then
P(A then B) = P(A) £ P(B given that A has occurred).
A typical example of dependent events is when we sample two objects without replacement.
This means that the first object is not replaced before the second is selected. It therefore cannot be selected twice.
Example 4
Self Tutor
A box contains 4 red and 2 yellow tickets. Two tickets are randomly selected one after the other from the box, without replacement. a Display this information on a tree diagram. b What is the probability that both are red? c What is the probability that one is red and the other is yellow? a Let R be the event that a red ticket is drawn and Y be the event that a yellow ticket is drawn.
Note that the outcome of the second event depends on the first.
1st selection
Ry_
R
Wy_
Y
2nd selection
Et_
R
Ry_ ´ Et_
Wt_
Y
Ry_ ´ Wt_
Rt_
R
Wy_ ´ Rt_
Qt_
Y
Wy_ ´ Qt_
b P(R and then R) = P(R) £ P(R given that R has occurred)
=
4
6
£
3
5
=
2
5
P(one is R and the other is Y )
= P(R and then Y or Y and then R)
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95
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75
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=
4
2
6 £ 5 +
16
or
30
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=
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IB MYP_4
408
TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
EXERCISE 20B.2
1 A box contains 7 red and 3 green balls. Two balls are randomly selected from the box one after the other. The first is not replaced in the box before the second is selected.
Determine the probability that: a both are red b the first is green and the second is red c a green and a red are obtained.
Selecting items simultaneously means there is no replacement.
2 A hat contains the names of the 7 players in a tennis squad including the captain and the vice captain. A team of 3 players is chosen at random by drawing the names from the hat. a Display this information on a tree diagram.
Distinguish between the captain, the vice captain, and the other players. b Find the probability that the team: i does not contain the captain ii contains neither the captain nor the vice captain iii contains either the captain or the vice captain, but not both.
3 Amelie has a bag containing two different varieties of apples. They are approximately the same size and shape, but one variety is red and the other is green. There are 4 red apples and 6 green ones. She selects one apple at random, eats it, and then takes another, also at random. Determine the probability that: a both apples were red b both apples were green c the first was red and the second was green d the first was green and the second was red e she ate one red and one green apple.
4 Marjut has a carton containing 10 cans of soup. 4 cans are tomato and the rest are pumpkin. She selects 2 cans at random without looking at the labels. a Let T represent tomato and P represent pumpkin. Draw a tree diagram to illustrate this sampling process. b What is the probability that both cans were tomato soup? c What is the probability that one can was tomato and the other can was pumpkin soup?
LARGE SAMPLE SPACES
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Sometimes the number of possible outcomes is sufficiently large that it is a waste of time to draw a tree diagram. We can still use the same principles we used with the tree diagrams to calculate the probabilities.
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IB MYP_4
TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
Example 5
409
Self Tutor
A hat contains 20 tickets with the numbers 1, 2, 3, ..., 19 and 20 printed on them.
If 3 tickets are drawn from the hat without replacement, determine the probability that all are prime numbers.
There are 20 numbers of which 8 are primes: f2, 3, 5, 7, 11, 13, 17, 19g
) P(3 primes)
= P(1st drawn is prime and 2nd is prime and 3rd is prime)
8
7
6
£ 19
£ 18
= 20
8 primes out of 20 numbers
7 primes out of 19 numbers after a successful first draw
6 primes out of 18 numbers after two successful draws
¼ 0:0491
EXERCISE 20B.3
1 A lottery has 100 tickets which are placed in a barrel. Two tickets are drawn at random from the barrel to decide 2 prizes. If John has 2 tickets in the lottery, determine his probability of winning: a first prize
b first and second prize
c second prize but not first prize.
d none of the prizes.
2 A bin contains 12 identically shaped chocolates of which 8 are strawberry creams. If 3 chocolates are selected at random from the bin, determine the probability that: a they are all strawberry creams
b none of them are strawberry creams.
3 A bag contains two white and five red marbles. Three marbles are selected simultaneously. Determine the probability that: a all are red
b only two are red
c at least two are red.
SELECTION WITH AND WITHOUT REPLACEMENT
You may have noticed that when we work with tree diagrams, the probabilities of independent and dependent events are calculated using the same method. The following example compares two such events.
Consider a box containing 3 red, 2 blue and 1 green marble. Suppose we wish to sample two marbles: ² with replacement of the first before the second is drawn
² without replacement of the first before the second is drawn.
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Examine how the tree diagrams differ:
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TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
With replacement
Without replacement
1st
Ey_
R
Wy_
2nd
R
B ( )
Qy_
Ey_
G
R ( )
Ey_
B
Wy_
Ey_
G
R
Qy_
Wy_
B
Qy_
G
Ey_
Wy_
Wy_
B
Qy_
Qy_
G
This branch represents Blue with the first draw and Red with the second draw. This is written as BR.
Notice that:
Wt_
R
Wt_
B ( )
Qt_
G
Et_
R ( )
Qt_
B
B
Qt_
Et_
Wt_
G
² with replacement
P(two reds) =
² without replacement
P(two reds) =
3
6
3
6
£
£
3
6
2
5
=
=
2nd
R
1st
G
R
G can’t have
GG
1
4
1
5
We can thus see why replacement is important.
Example 6
Self Tutor
For the example above of the box containing 3 red, 2 blue and 1 green marble, find the probability of getting a red and a blue:
a with replacement
b without replacement.
a
b
P(a red and a blue)
= P(RB or BR)
=
=
=
3
2
6 £ 6
12
36
1
3
+
2
6
£
P(a red and a blue)
fticked onesg
3
6
= P(RB or BR)
3
2
6 £ 5
12
30
2
5
=
=
=
+
2
6
£
3
5
fcrossed onesg
EXERCISE 20B.4
Use tree diagrams to help answer the following questions:
1 Jar A contains 4 blue and 2 red marbles. Jar B contains 1 blue and 5 red marbles. A jar is randomly selected and one marble is taken from it. Determine the probability that the marble is blue.
2 Two marbles are drawn in succession from a box containing 2 purple and 5 green marbles.
Determine the probability that the two marbles are different colours if: a the first is replaced
b the first is not replaced.
3 5 tickets numbered 1, 2, 3, 4 and 5 are placed in a bag. Two are taken from the bag without replacement. Determine the probability that:
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c one is odd and the other is even.
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TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
4 Jar A contains 3 red and 2 green tickets. Jar B contains 3 red and
7 green tickets. A die has 4 faces marked A and 2 faces marked B.
The die is rolled and the result is used to select either jar A or jar B.
When a jar has been selected, two tickets are randomly selected from it without replacement. Determine the probability that: a both are green
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A
A
B
b they are different in colour.
Example 7
Self Tutor
A bag contains 5 red and 3 blue marbles. Two marbles are drawn simultaneously from the bag. Determine the probability that at least one is red.
P(at least one red) = P(RR or RB or BR)
draw 2 draw 1
Ru_
R
=
Eu_
B
=
Tu_
R
Wu_
B
R
Ti_
Ei_
=
=
B
5
4
5
8 £ 7 + 8
20+15+15
56
50
56
25
28
£
3
7
+
3
8
£
5
7
Alternatively, P(at least one red) = 1 ¡ P(no reds) fcomplementary eventsg
= 1 ¡ P(BB) and so on.
5 A bag contains four red and two blue marbles. Three marbles are selected simultaneously.
Determine the probability that: a all are red
b only two are red
c at least two are red.
6 Box A contains 3 red and 4 green marbles.
Box B contains 5 red and 2 green marbles.
One marble is randomly selected from box A and
A
its colour noted. If it is red, 2 reds are added to box B. If it is green, 2 greens are added to box B.
B
A marble is then selected from box B.
Find the probability that the marble selected from box B is green.
C
BINOMIAL PROBABILITIES
In many situations there are only two possible outcomes. For example:
² a tennis player either wins or loses a game
² you either catch a bus or you don’t
² you either make a free throw or you miss.
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Binomial experiments are concerned with the repetition of several independent trials where there are only two possible outcomes, success and failure. Each trial has the same probability of success.
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