Conditional Probability and the Multiplication Rule Section 4.3 Conditional Probability and the Multiplication Rule

Definitions Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F. Two events are dependent if the occurrence of event E in a probability experiment affects the probability of event F.

Multiplication Rule (Independent Events) Note: Can have more than two events

1. Using the formula E and F are independent. If P(E) = 0.5 and P(F) = 0.3. What is the P(E and F)?

Notation P(A and B) = P(event A occurs in a 1st trial and event B occurs in a 2nd trial)

2. What is the probability What is the probability of rolling a die and getting a 5 and flipping a coin and getting heads?

3. What is the probability Given the probability that you are going to die from cancer (versus other causes) is 0.23, what is the probability that four people picked at random are going to die from cancer?

4. What is the probability Given a standard deck of cards and we pick 3 cards out with replacement, what is the probability of getting 3 kings?

5. What is the probability We are rolling a pair of dice, what is the probability that we get a total of seven (counting the dots) 4 times in a row?

At-Least One Probabilities Form: Find probability of at least one {something} out of “n” Write as: 1-P(none {something}) Write as: 1-P(all {opposite of something}) Write as: 1-P(all {opposite of something})n

“At Least One” Rule & Complements Find probability of none happening and then subtract from 1 Example: Out of 5 births, find the probability of at least getting one girl. P(at least one girl in 5 births) =1-P(no girls in 5 births) =1-P(all boys in 5 births) =1-(1/2)(1/2)(1/2)(1/2)(1/2)

6. Find the probability The test contains 25 multiple choice problems each with 4 parts, only one of which is right. You are guessing, what is the probability you will get at least one right?

7. Find the probability The odds you will die from fireworks is 1 in 652,046. What is the probability that out of 5 people picked at random, at least one will die from fireworks?

Definition Conditional Probability = The notation P(F|E) is read “the probability of event F given event E”. It is the probability that the event F occurs, given that the event E has occured

Conditional Probability Rule The probability of event F occurring, given the occurrence of event E, is found by dividing the probability of E and F by the probability of E, or by dividing the number of outcomes in E and F by the number of outcomes in E.

8. Suppose that E and F are two events and that P(E and F) = 0 8. Suppose that E and F are two events and that P(E and F) = 0.5 and P(E) = 0.7. What is P(F|E)?

9. Suppose that E and F are two events and that N(E and F) = 300 and N(E) = 800. What is P(F|E)?

10. Suppose that E and F are two events and that P(E) = 0 10. Suppose that E and F are two events and that P(E) = 0.7 and that P(F|E) = 0.3. What is P(E and F)?

11. Given the general population, 12% are old 11. Given the general population, 12% are old. In addition 8% are old and forgetful. What is the probability that a randomly selected person will be forgetful given they are old?

12. For 18-20 year olds, 27% believe in aliens and ghosts 12. For 18-20 year olds, 27% believe in aliens and ghosts. Given that 35% of 18-20 years believe in ghosts given they believe in aliens, what probability believe in aliens?

13. Given the following table Male Female Total Brunette 25 10 35 Red 2 7 9 Blonde 5 15 20 32 64 What is the probability that a randomly selected person is female given they have red hair? What is the probability that a randomly selected person has blonde hair given they are male?

General Multiplication Rule (Dependent Events)

14. Find the probability Suppose that three cards are randomly selected from a standard deck of cards: What is the probability of getting three queens with replacement? What is the probability of getting three queens with no replacement?

15. Find the probability There are ten men and thirteen women in a stats class. Two are to selected to attend a statistic conference. What is the probability that if two are selected at random, that they are both women?

16. Find the probability Given a standard deck of cards, what is the probability of getting 3 of a kind (3 two’s, 3 queen’s etc.)? An example hand: Q-Q-Q-2-5 (we cannot have 4 queens since that would be 4 of a kind). Note: (assuming you are dealt the first five cards and the first 3 cards are 3 of the same)

17. Find the probability In a box, we have 10 black markers, 5 blue markers, 3 red markers and 7 green markers. We are choosing two markers at random: What is the probability they are both green? What is the probability the first is green and the second one is black? What is the probability the first is black and the second one is green? What is the probability the one is black and one is green?

Note If small random samples are taken from large populations without replacement, it is reasonable to assume independence of the events. As a rule of thumb, if the sample size is less than 5% of the population size, we treat the events as independent.

Definition Two events E and F are independent if P(E|F) = P(E) or, equivalently, if P(F|E) = P(F)