Math 310 Section 7.4 Probability

Odds Def Let P(A) be the probability that A occurs and P(Ā) be the probability that A does not occur. Then the odds in favor of an event A are: P(A)/P(Ā) or P(A)/(1-P(A)) The odds against are P(Ā)/P(A).

Odds again In terms of equally likely outcomes: Odds in favor = Number of favorable outcomes Number of unfavorable outcomes Number of unfavorable outcomes Odds against = Number of unfavorable outcomes Number of favorable outcomes Number of favorable outcomes

Notation Often a colon is used instead of a fraction: Odds in favor 4/3 or 4:3 Odds against 3/4 or 3:4

Probability from odds Thrm If the odds in favor of event E are m : n then P(E) = m/(m + n) If the odds against E are m : n then P(E) = n/(m + n) P(E) = n/(m + n)

Ex. What are the odds in favor of rolling a number greater than 4 on a die? 2 : 4 or 1:2

Ex. What are the odds of throwing a tails on a penny? 1 : 1

Ex. What are the odds of drawing a 2 or a 3 from a 52 card deck? 8 : 44 or 2 : 11

Conditional Probability Conditional probability is the probability of a sequence of events occurring given that one or more events in the sequence is guaranteed to occur.

Conditional Probability Thrm If A and B are events in a sample space S and P(A) ≠ 0, then the conditional probability that event B occurs given that event A has occurred is given by P(B|A) = P(A ∩ B) P(A) P(A)

The Idea The basic idea for conditional probability is that by adding more information we reduce the sample space.

Ex. What is the probability of rolling a 2 if you know the roll is a prime number? S = {1, 2, 3, 4, 5, 6}, A = {2} But if we know the roll is a prime then S * = {2, 3, 5}. So the P(A) = 1/3