1. Independent Events Slideshow 54, Mathematics Mr Richard Sasaki, Room 307

2. Objectives Recalling the meaning of “with replacement” and “without replacement” Understand independence and calculating probabilities about 2 events

3. Vocabulary We need a bit of a review. Event (Trial) - The thing that is taking place (eg: Rolling a die) Value -Possible outcomes for the event (for a die: 1, 2, 3, 4, 5, 6) Frequency - The number of times a value appears in an experiment.

4. Independence In the Winter Homework, Independence was mentioned. What is it again? Independence for events is where one event doesn’t affect another. This means that no matter what happens in one event, the probabilities for the other event are exactly the same. Last lesson we looked at pulling objects out of a bag and “with replacement” and “without replacement”. Let’s review those meanings.

5. With and Without Replacement With Replacement – After an event occurs, everything is “reset” (put back as it was) so when we repeat, nothing has changed. Without Replacement – After an event occurs, whatever happened is removed from the event, causing all future occurrences to have differing probabilities. Which of these shows independence? With Replacement

6. # of outcomes Flipping a Coin…Twice! Let’s consider flipping an unbiased coin twice. What are the possibilities we can get? Let’s list them… ① Heads, Heads ② Heads, Tails ③ Tails, Heads ④ Tails, Tails Note – Heads, Tails is different to Tails, Heads. When listing possible outcomes, order does have meaning. Probability (of happening) Why is each ¼? # of successes There are 4 combinations.

7. Flipping a Coin…Twice! How about the following? P(A Heads and a Tails) = Order isn’t mentioned. P(H and T) = P(H, T) + P(T, H). P(A Heads or a Tails) = We always get heads or tails! The terms “and”, “or” and “,” are all very different when there are 2 or more events taking place. And- Both must happen (any order) Or - At least one of them must happen, - Both must happen in the given order

8. Answers - Easy P(Tails) = ½ Yes, their outcomes don’t affect each other. P(H, T) = ¼ No. Both events are independent with the same probabilities for each outcome. 1, 2, 3 1, 12, 13, 1 1, 22, 23, 2 1, 32, 33, 3

9. Answers - Hard 16 12 P(Two sixes) = 0 Because order is considered. The coin is flipped first so we can’t flip a 3.

10. An Introduction to Permutations Two options, two picked (Coin) - 4 Three options, two picked (Spinner) - 9 Four options, two picked - 16 Six options, two picked - 36 Try the discovery worksheet about permutations!

11. An Introduction to Permutations Permutations are combinations where order matters. (These are like the ones we did today.) So if we rolled a 10 sided die four times, how many permutations exist?