1. Chapter 22 E

2. Outcomes of Different Events When the outcome of one event affects the outcome of a second event, we say that the events are dependent. When one outcome of one event does not affect a second event, we say that the events are independent.

3. Classify each pair of events as dependent or independent. Probability of Multiple Events a.Spin a spinner. Select a marble from a bag that contains marbles of different colors. Since the two events do not affect each other, they are independent. b.Select a marble from a bag that contains marbles of two colors. Put the marble aside, and select a second marble from the bag. Picking the first marble affects the possible outcome of picking the second marble. So the events are dependent.

4. Decide if the following are dependent or independent An expo marker is picked at random from a box and then replaced. A second marker is then grabbed at random. Two dice are rolled at the same time. An Ace is picked from a deck of cards. Without replacing it, a Jack is picked from the deck. Independent Dependent

5. How to find the Probability of Two Independent Events If A and B are independent events, the P(A and B) = P(A) ● P(B)  Ex: If P(A) = ½ and P(B) = 1/3 then P(A and B) =

6. Example: Finding the Probability of Independent Events Tossing red, then white, then yellow. The result of any toss does not affect the probability of any other outcome. 4 of the 6 sides are red; 1 is white; 1 is yellow. A 6-sided die has 4 red sides, one side is white, and one side is yellow. Find the probability. P(red, then white, and then yellow) = P(red)  P(white)  P(yellow)

7. Events are dependent events if the occurrence of one event affects the probability of the other. For example, suppose that there are 2 lemons and 1 lime in a bag. If you pull out two pieces of fruit, the probabilities change depending on the outcome of the first.

8. The tree diagram shows the probabilities for choosing two pieces of fruit from a bag containing 2 lemons and 1 lime. The probability of a specific event can be found by multiplying the probabilities on the branches that make up the event. For example, the probability of drawing two lemons is

9. Homework Responsible for ALL of section E (both E1 & E2)

10. Chapter 22 G – Sampling With & Without Replacement

11. With Replacement – Coin tosses, dice, Roulette, and DNA. “ memoryless ” – After you get heads, you have an equally likely chance of getting a heads on the next toss (unlike in Poker, where you can’t draw the same card twice from a single deck). What’s the probability of getting two heads in a row (“HH”) when tossing a coin? H H T T H T Toss 1: 2 outcomes Toss 2: 2 outcomes 2 2 total possible outcomes: {HH, HT, TH, TT} With Replacement

12. What’s the probability of 3 heads in a row? With Replacement H H T T H T Toss 1: 2 outcomes Toss 2: 2 outcomes Toss 3: 2 outcomes H T H T H T H T HHH HHT HTH HTT THH THT TTH TTT

13. WITH REPLACEMENT — You have a bag containing 3 red, 2 blue, and 1 yellow moist socks. What is P(B & R) – In order? PUT the first sock back in the bag

14. Without Replacement Without replacement — Think cards (w/o reshuffling) and seating arrangements. Example: You have a bag containing 3 red, 2 blue, and 1 yellow moist socks. What’s the probability we pull 1 blue AND 1 red if we don’t put the first sock back in the bag?

15. WithOUT Replacement WithOUT —You have a bag containing 3 red, 2 blue, and 1 yellow moist socks. What is P(B & R)? DON’T PUT the first sock back in the bag

16. Homework Page 589 (1 – 10) I would pay close attention to #10