Chapter 3 Probability 3.2 Conditional Probability and the Multiplication Rule Conditional Probability A conditional probability is the probability of an event occurring, given that another event has already occurred. P (B | Probability of B, given A A) Example: There are 5 red chip, 4 blue chips, and 6 white chips in a basket. Two chips are randomly selected. Find the probability that the second chip is red given that the first chip is blue. (Assume that the first chip is not replaced.)
Because the first chip is selected and not replaced, there are only 14 chips remaining. 5 0.357 P (selecting a red chip|first chip is 14 blue) Larson & Farber, Elementary Statistics: Picturing the World, 3e 3 Conditional Probability Example: 100 college students were surveyed and asked how many hours a week they spent studying. The results are in the table below. Find the probability that a student spends more than 10 hours studying given that the student is a male. Less More Male Female Total
then 5 11 13 24 5 to 10 than 10 22 16 24 14 46 30 Total 49 51 100 The sample space consists of the 49 male students. Of these 49, 16 spend more than 10 hours a week 16 studying.
0.327 P (more than 10 hours|male) 49 Larson & Farber, Elementary Statistics: Picturing the World, 3e 4 Independent Events Two events are independent if the occurrence of one of the events does not affect the probability of the other event. Two events A and B are independent if P (B |A) = P (B) or if P (A |B) = P (A). Events that are not independent are dependent. Example: Decide if the events are independent or dependent. Selecting a diamond from a standard deck of cards (A), putting it back in the deck, and then selecting a spade from the deck13 (B).1 The occurrence of A does 13 1
P (B A ) 52 4 and P (B ) 52 . 4 not affect the probability of B, so the events are independent. Larson & Farber, Elementary Statistics: Picturing the World, 3e 5
Multiplication Rule The probability that two events, A and B will occur in sequence is P (A and B) = P (A) P (B |A). If event A and B are independent, then the rule can be simplified to P (A and B) = P (A) P (B). Example: Two cards are selected, without replacement, from a deck. Find the probability of selecting a diamond, and then selecting a spade. Because the card is not replaced, the events are dependent. P (diamond and spade) = P (diamond) P (spade |diamond) 13 13 169 0.064 52 51 2652 Larson & Farber, Elementary Statistics: Picturing the World, 3e 6 Multiplication Rule Example:
A die is rolled and two coins are tossed. Find the probability of rolling a 5, and flipping two tails. 1 P (rolling a 5) = . 6 1 Whether or not the roll is a 5, P (Tail ) =, 2 so the events are independent. P (5 and T and T ) = P (5) P (T ) P (T ) 1 1 1 6 2 2 1 0.042 24 Larson & Farber, Elementary Statistics: Picturing the World, 3e 7