Independence and Conditional Probability

Warm-Up If P(A) = 0.3 and P(B) = 0.4 and if A and B are mutually exclusive events, find: a. b. c. d. a. 0.7 b. 0.6 c. 0.7 d. 0

Multiplication Rule – Independent Events…… When 2 events are independent, the probability of both occurring is

General Rule…… “or” means to add “and” means to multiply (unless it is in a contingency table and you can actually see the intersection)

Example…… If a coin is tossed twice, find the probability of getting 2 heads. Answer:

Example…… A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die. Answer:

Example…… A card is drawn from a deck and replaced; then a 2nd card is drawn. Find the probability of getting a queen and then an ace. Answer:

Example…… A box contains 3 red balls, 2 blue balls, and 5 white balls. A ball is selected and its color noted. Then it is replaced. A 2nd ball is selected and its color noted. Find the probability of Selecting 2 blue balls Selecting a blue ball and then a white ball Selecting a red ball and then a blue ball

Answers…… Selecting 2 blue balls Selecting a blue ball and then a white ball Selecting a red ball and then a blue ball

Example…… A poll found that 46% of Americans say they suffer from stress. If 3 people are selected at random, find the probability that all three will say they suffer from stress. Answer:

Dependent Events…… When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed.

Examples of Dependent Events…… Draw a card from a deck. Do not replace it and draw another card. Having high grades and getting a scholarship Parking in a no parking zone and getting a ticket

Multiplication Rule – Dependent Events…… When 2 events are dependent, the probability of both occurring is The slash reads: “The probability that B occurs given that A has already occurred.”

Example…… 53% of residents had homeowner’s insurance. Of these, 27% also had car insurance. If a resident is selected at random, find the prob. That the resident has both homeowner’s and car insurance. Answer:

Example…… 3 cards are drawn from a deck and NOT replaced. Find the following probabilities. a. Getting 3 jacks b. Getting an ace, king, and queen c. Getting a club, spade, and heart d. Getting 3 clubs.

a. Getting 3 jacks……

b. Getting an ace, king, queen……

c. Getting a club, spade, and heart……

d. Getting 3 clubs……

Dependent Probability Continued……Conditional

Warm Up……How Likely Are You to Win the Lotto? Many states have lotteries. The biggest jackpot, typically millions of dollars, usually comes from the Lotto game. In Lotto South, available in Georgia, Kentucky, and Virginia, six numbers are randomly sampled without replacement from the integers 1 to 49. The order of selection is not important. Question: You buy a lottery ticket. What is the probability that it is a winning ticket, having the six numbers chosen?

The probability of winning is the probability that the 6 numbers chosen are the six that you have on your ticket. Keep in mind that the order does not matter and that a number cannot be repeated after it has been chosen. Find the probability of winning.

Answer…… This is about 1 chance in 14 million!

Insight……Provided by Wilson and Crouch 2001, p. 200 Let’s give this small number some perspective. The chance of winning the jackpot in Lotto South (0.00000007) is…… less than your chance of being hit by a meteorite in the next year (0.0000004). less than your chance of dying in a tornado (0.0000002). less than your chance of dying by a lightning strike (0.00000016).

In other words…… If you have money to spare, go ahead and play the lottery, but understand why many call it “sport for the mathematically challenged.” By the way, the probability of winning the Lotto South is also roughly the probability that a person of average mortality will die in the next 3 minutes! Do you still want to play?

Back to Conditional Probability - Remember…… Algebraically change this so that it is now in the form……

“Given”

Example…… In Rolling Acres Housing Plan, 42% of the houses have a deck and a garage; 60% have a deck. Find the probability that a home has a garage, given that it has a deck.

Answer…… Answer:

Example…… At an exclusive country club, 83% of the members play bridge; 75% of the members drink champagne given that he or she plays bridge. Find the probability that members drink champagne and play bridge.

Answer…… Answer:

Example…… A recent survey asked 100 people if they thought women in the armed forces should be permitted to participate in combat. The results are shown in the table. Yes No Total Male 32 18 50 Female 8 42 40 60 100

a. Find the probability that they answered yes, given that they were female. No Total Male 32 18 50 Female 8 42 40 60 100 Answer:

b. Find the probability that they were male, given that they answered no. Yes No Total Male 32 18 50 Female 8 42 40 60 100