Independent & Dependent Events

Review How can you tell whether events are Bell Ringer: Mixed review Review How can you tell whether events are mutually exclusive (disjoint)? When two or more events are mutually exclusive, how do you find the probability of one event or the other happening?

Review (cont’d) What is the Counting Principle? When and how it is used? How do you use a tree diagram?

Independent Events Independent events are two or more events whose outcomes do not affect one another They can happen at the same time, however Example: The BMS Booster Club is selling 150 raffle tickets for a gift card to the movies. Y Club is also selling 200 raffle tickets for free pizza. You buy 5 raffle tickets from both clubs. What is the probability that you win both prizes?

Independent Events Example These two raffle drawings are independent events because the outcome of the Booster Club drawing has no effect on the Chess Club drawing In order to calculate the probability of winning both drawings, we must first find the probability of winning each one separately

Independent Events Example (cont’d) Let “winning the BMS drawing” be Event A Let “winning the Y Club drawing” be Event B P(A) = P(B) = Now, we want to find the probability of winning both P(A and B)

Example (slide 3) P(A) = P(B) = To find P(A and B), we must multiply the probabilities together P(A and B) =

Example (slide 4) Hence, the probability of winning both the Booster Club raffle and the Y Club raffle is about 1 in 1,200, or 0.083%.

Dependent Events Dependent events are two or more events where the outcome of one does affect the outcome of the other Example: From a standard deck of cards, what is the probability of drawing a queen, then without replacement, drawing another queen?

Dependent Events Example Without replacement is a very important phrase This means that when the first card is drawn, it is not put back in the deck before drawing the second “Without replacement” means that the events are dependent Events are dependent when the sample size changes from one event to the next

Dependent Events Example (cont’d) In this situation, the sample size has changed from 52 cards to 51, since the first card was not put back in the deck Let’s say that the probability of drawing the first queen is Event A, and the probability of drawing the second queen is Event B

Dependent Events Example (slide 3) Just as with independent events, we also need to first find P(A) and P(B|A) separately before finding P(A and B) P(B|A) means “probability of B after A” We must take A into consideration when calculating P(B)

Dependent Events Example (slide 4)

Closure Distinguish between independent and dependent events BR tomorrow: coin flipping w/ multiple coins Distinguish between independent and dependent events Explain how to find P(A and B) for independent events Explain how to find P(A and B) for dependent events