Random Variable and Expected Average Return
APStatistics Cole Rogers
Unit 7 Exam Random Variables: Free Response
Directions: Complete the assignment on this paper. If you need additional paper make sure that you clearly label each page with your name. Your answers for this assignment must include reasons; simply stating the answer without justification will earn partial credit.
1. A Roulette wheel has 38 slots numbered 0 to 36 and 00. The wheel is spun and a ball is thrown into the wheel and comes to rest in one of the slots. There are numerous of ways to bet, individual numbers, groups of numbers (1-12, 13-24, etc), by color (half of the numbers are black and the half are red), and in various other combinations. This problem is going to focus on betting $1.00 on the number group 1-12. If the ball lands in any of the values 1-12 the bet is won and the return is $3.00. If the ball lands on any of the other values the bet is lost.
a.) Compute the expected value of this game. (4 points)
X
2
-1
P(x)
12/38
26/38
X*P(x)
.631
-.684
-.053
The expected value is -.053.
b.) Interpret this expected value. (4 points)
This means that the casino wins 5.3 cents per game or the players lose 5.3 cents per game.
c.) What is the average return to the casino from 1,000,000 such bets? (4 points)
1000000*.05=50000
The casino can expect to win 500000 of the 1 million bets because 1 million times .05 is 50,000.
2. A venture capital fund has the mandate to invest in new businesses that may be perceived as being too risky by other investors. Suppose that a fund invests its funds in units with no more than one unit per firm in order to diversify its investments. Furthermore, suppose there are two classes of firms with the following distributions of net returns, and assume that all firms operate independently of each other.
Type A firms
Type B firms
% returns
-15
0
45
-25
-5
0
25
65
Probability
.30
.30
.40
.15
.15
.30
.20
.20
Unit 7 Exam Random Variables: Free Response
Directions: Complete the assignment on this paper. If you need additional paper make sure that you clearly label each page with your name. Your answers for this assignment must include reasons; simply stating the answer without justification will earn partial credit.
1. A Roulette wheel has 38 slots numbered 0 to 36 and 00. The wheel is spun and a ball is thrown into the wheel and comes to rest in one of the slots. There are numerous of ways to bet, individual numbers, groups of numbers (1-12, 13-24, etc), by color (half of the numbers are black and the half are red), and in various other combinations. This problem is going to focus on betting $1.00 on the number group 1-12. If the ball lands in any of the values 1-12 the bet is won and the return is $3.00. If the ball lands on any of the other values the bet is lost.
a.) Compute the expected value of this game. (4 points)
X
2
-1
P(x)
12/38
26/38
X*P(x)
.631
-.684
-.053
The expected value is -.053.
b.) Interpret this expected value. (4 points)
This means that the casino wins 5.3 cents per game or the players lose 5.3 cents per game.
c.) What is the average return to the casino from 1,000,000 such bets? (4 points)
1000000*.05=50000
The casino can expect to win 500000 of the 1 million bets because 1 million times .05 is 50,000.
2. A venture capital fund has the mandate to invest in new businesses that may be perceived as being too risky by other investors. Suppose that a fund invests its funds in units with no more than one unit per firm in order to diversify its investments. Furthermore, suppose there are two classes of firms with the following distributions of net returns, and assume that all firms operate independently of each other.
Type A firms
Type B firms
% returns
-15
0
45
-25
-5
0
25
65
Probability
.30
.30
.40
.15
.15
.30
.20
.20