Section 5.3 Independence and the Multiplication Rule

Section 5.3 – Independence and the Multiplication Rule Objectives 1. Identify independent events 2. Use the Multiplication Rule for Independent Events 3. Compute at-least probabilities

Independent Events – Events E and F are independent if the fact that E occurs does not affect the probability of F occurring.

Dependent Events - Two events are dependent if the outcome or occurrence of the first affects the outcome or occurrence of the second so that the probability is changed.

Put another way, when two events are dependent, the occurrence of one event influences the probability of the other event.

Disjoint events are dependent.

Simple examples of independent events:  Buying a lottery ticket and finding a penny on the floor (your odds of finding a penny does not depend on you buying a lottery ticket).  Taking a cab home and finding your favorite movie on cable.  Getting a parking ticket and playing craps at the casino.

Simple examples of dependent events:  Parking illegally and getting a parking ticket. Parking illegally increases your odds of getting a ticket.  Buying ten lottery tickets and winning the lottery. The more tickets you buy, the greater your odds of winning.  Driving a car and getting in a traffic accident.

Example: Create the sample space for the following scenario:

Answer two questions on a test. Question #1 is true/false and Question #2 is multiple choice (3 choices).

1 What is the probability your answers are “T” and “C” ? P(true and C) =

Notice that P(T) = ______and P(C) = ______

The Multiplication Rule for Independent Events: P(E and F) = P(E) · P(F)

Example: According to the National Vital Statistics Report, the probability that a randomly selected 24-year old male will survive the year is 0.9986.

a) What is the probability that 2 randomly selected 24-year-old males will survive the year?

b) What is the probability that 5 randomly selected 24-year-old males will survive the year?

Example: A Roulette wheel contains 38 slots. You bet on the number 5. What is the probability that the ball will land on slot #5 two times in a row?

Example: A bag contains 3 blue marbles, 2 red marbles, and 4 white marbles. You draw a marble and put it back before drawing another marble. (i.e., with replacement) a) P(1st is red and 2nd is blue)

b) P(1st is red and 2nd is blue and 3rd is blue)

2 “With replacement” makes the events independent “No replacement” makes the events dependent

Example: A bag contains 3 blue marbles, 2 red marbles, and 4 white marbles. You draw a marble and don’t put the marble back before drawing another marble. (i.e., no replacement) a) P(1st is red and 2nd is blue)

b) P(1st is red and 2nd is blue and 3rd is blue)

Computing “At-Least” Probabilities:

The opposite (complement) of “at least one” is “less than one.”

P(at least one) = 1 – P(less than one)

Example: Flip a coin 3 times. S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} P(no heads) =

P(at least one head) =

Example: Every day, George buys a lottery ticket. For each ticket, the probability of winning a prize is 0.2. After seven days, what is the probability that George has won at least one prize?

3 Section 5.4 – Conditional Probability and the General Multiplication Rule Objectives 1. Compute conditional probabilities 2. Compute probabilities using the General Multiplication Rule

Conditional Probability – The probability of F occurring given that event E has already occurred is denoted as P(F│E) and is read “the probability of F given E”

Example : Suppose a single die is rolled. Calculate the following probabilities: S= {1, 2, 3, 4, 5, 6} a) P(roll a 3 | roll an odd) =

b) P(roll a 2 | roll an odd) =

c) P(roll a 1 | roll a number less than 5) =

d) P(roll a 6 | roll a number that is a multiple of 3) =

Example: Roll a single fair die one time and define the following events: E = {2,4,6} L = {1,2} H={5,6} A={1,2,3,4} Find the following probabilities: Remember the sample space is S={1,2,3,4,5,6}

P(E) = P(L) =

P(H) = P(A) =

P(L|E) = P(L|A) =

P(L|H) = P(E|A) =

4 Example: You choose 3 cards one at a time from a standard deck of cards. Calculate the following probabilities: a) P(all 3 cards are Kings) = (sampling done with replacement)

b) P(all 3 cards are Kings) = (sampling done without replacement)

5 Example: (pg.273)

If ONE person is randomly selected from the above survey, calculate the following probabilities: a) P(male) =

b) P(divorced) =

c) P(female│married) =

d) P(never married│male) =

e) P(male│separated) =

6 Example: A Bond Referendum – An elementary school bond referendum which requires a majority vote to pass is put before the voters in a school district where 30% of the voters have children in elementary school and 70% do not. Polls indicate that 90% of those with children in elementary school will vote for the bond issue, while only 205 of those without children in elementary school will do so. If the polls are right, will the bond issue pass?