Events
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Stats: Probability Rules
"OR" or Unions
Mutually Exclusive Events
Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is disjoint.
If two events are disjoint, then the probability of them both occurring at the same time is 0. Disjoint: P (A and B) = 0
If two events are mutually exclusive, then the probability of either occurring is the sum of the probabilities of each occurring.
Specific Addition Rule
Only valid when the events are mutually exclusive. P (A or B) = P (A) + P (B)
Example 1:
Given: P(A) = 0.20, P(B) = 0.70, A and B are disjoint
I like to use what's called a joint probability distribution. (Since disjoint means nothing in common, joint is what they have in common -- so the values that go on the inside portion of the table are the intersections or "and"s of each pair of events). "Marginal" is another word for totals -- it's called marginal because they appear in the margins. | B | B' | Marginal | A | 0.00 | 0.20 | 0.20 | A' | 0.70 | 0.10 | 0.80 | Marginal | 0.70 | 0.30 | 1.00 |
The values in red are given in the problem. The grand total is always 1.00. The rest of the values are obtained by addition and subtraction.
Non-Mutually Exclusive Events
In events which aren't mutually exclusive, there is some overlap. When P(A) and P(B) are added, the probability of the intersection (and) is added twice. To compensate for that double addition, the intersection needs to be subtracted.
General Addition Rule
Always valid. P (A or B) = P (A) + P (B) - P (A and B)
Example 2:
Given P(A) = 0.20, P(B) = 0.70, P(A and B) = 0.15 | B | B' | Marginal | A | 0.15 | 0.05 | 0.20 | A' | 0.55 | 0.25 | 0.80 | Marginal | 0.70 | 0.30 | 1.00 |
Interpreting the table
Certain things can be determined from the joint probability distribution. Mutually exclusive events will have
Stats: Probability Rules
"OR" or Unions
Mutually Exclusive Events
Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is disjoint.
If two events are disjoint, then the probability of them both occurring at the same time is 0. Disjoint: P (A and B) = 0
If two events are mutually exclusive, then the probability of either occurring is the sum of the probabilities of each occurring.
Specific Addition Rule
Only valid when the events are mutually exclusive. P (A or B) = P (A) + P (B)
Example 1:
Given: P(A) = 0.20, P(B) = 0.70, A and B are disjoint
I like to use what's called a joint probability distribution. (Since disjoint means nothing in common, joint is what they have in common -- so the values that go on the inside portion of the table are the intersections or "and"s of each pair of events). "Marginal" is another word for totals -- it's called marginal because they appear in the margins. | B | B' | Marginal | A | 0.00 | 0.20 | 0.20 | A' | 0.70 | 0.10 | 0.80 | Marginal | 0.70 | 0.30 | 1.00 |
The values in red are given in the problem. The grand total is always 1.00. The rest of the values are obtained by addition and subtraction.
Non-Mutually Exclusive Events
In events which aren't mutually exclusive, there is some overlap. When P(A) and P(B) are added, the probability of the intersection (and) is added twice. To compensate for that double addition, the intersection needs to be subtracted.
General Addition Rule
Always valid. P (A or B) = P (A) + P (B) - P (A and B)
Example 2:
Given P(A) = 0.20, P(B) = 0.70, P(A and B) = 0.15 | B | B' | Marginal | A | 0.15 | 0.05 | 0.20 | A' | 0.55 | 0.25 | 0.80 | Marginal | 0.70 | 0.30 | 1.00 |
Interpreting the table
Certain things can be determined from the joint probability distribution. Mutually exclusive events will have