(a) Suppose you flip a coin and roll a die. Are the events obtain a head and roll a 5 dependent or independent events?

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1 Unit 6 Probability Name: Date: Hour: Multiplication Rule of Probability By the end of this lesson, you will be able to Understand Independence Use the Multiplication Rule for independent events Independent events: Dependent events: Example 1: (a) Suppose you flip a coin and roll a die. Are the events obtain a head and roll a 5 dependent or independent events? (b) Suppose you draw one card from the deck and then replace it and draw a second card from the deck. Are the events getting a four the first time and an Ace the second time dependent or independent events? (c) Given event A is you are late for work and event B is your car has a flat tire, determine if events A & B are dependent or independent events. (d) Suppose you draw one card from the deck and then without replacing it, draw a second card from the deck. Are the events getting a heart the first time and a diamond the second time dependent or independent events?

2 Disjoint (Mutually Exclusive) Events vs. Independent Events **It is important that you understand that these are different concepts. ~ Two events are DISJOINT if they have. That is, if knowing one of the events occurs, we know the other did not occur. ~ INDEPENDENCE means that one event occurring the other event occurring. That is, knowing the outcome of one event does not mean we know anything about the second event (They can both still happen) Example 2: Suppose you flip a coin twice. What is the probability that you obtain a head on the first flip AND you obtain a head on the second flip? What is the sample space for this situation? From the sample space, determine the probability of heads twice. In this situation, writing the sample space was fairly easy. But, that is not always the case which is why we use the multiplication rule. Multiplication Rule for Independent Events Given two independent events E and F, Multiplication Rule for Dependent Events Given two dependent events E and F,

3 Example 3: You draw 1 card from a deck of cards, put it back, and the cards are shuffled. You draw three more cards the same way. What is the probability that you will draw 4 aces in a row? Example 4: You draw one card from a deck of cards, do not replace it, and draw a second card. What is the probability that the first card is an ace and the second card is a King? Example 5: The probability that a randomly selected student is taking an art class is 15%. What is the probability that three randomly selected students are taking an art class? Summary of Different Probability Situations AND Probability Two different events occurring one after another: First I draw a King, AND THEN, I draw a two. **You do an experiment OR you do different experiments.** OR Probability Two different events: one or the other can occur: I draw one card and it is a King OR a Queen. **You do an experiment, you are looking for outcomes.**

4 Quick Check Multiplication Rule of Probability 1. Determine if the following events are MUTUALLY EXCLUSIVE (DISJOINT) or INDEPENDENT. a. You roll a die once: Event E = You roll a 5 on a die. Event F = You roll a 3 on a die. b. You roll a die twice: Event E = You roll a 6 on a die. Event F = You roll a 2 on a die. 2. Determine if the following events are INDEPENDENT or DEPENDENT. a. You pick a card from the deck, do not put it back, and then pick another card. b. You pick a card from the deck, replace it, and then pick another card. 3. What is the probability of flipping a coin 5 times and getting heads 5 times in a row? (i.e. getting heads AND heads AND heads AND heads AND heads). Learning Goals Understand Independence Use the Multiplication Rule for independent events Self-Assessment I am unsure of or confused about this I am ready to start practicing I am already good at this My Goals for Today- thinking about what I am good at, where am I confused and what do I need to work on? What do I do if I am confused or need to work on a learning target?

5 Name: Date: Hour: Unit 6 Probability Multiplication Rule of Probability Homework 1. What is the probability of obtaining four ones in a row when rolling a fair, six-sided die four times? 2. The probability that someone is left-handed is If two people are randomly selected, what is the probability that both are left-handed? 3. The probability that a randomly selected 40-year-old male will live to be 41 years old is according to the National Vital Statistics Report, Vol. 48, No. 18. a. What is the probability that two randomly selected 40-year-old males will live to be 41 years old? b. What is the probability that five randomly selected 40-year-old males will live to be 41 years old? 4. You draw one card from a deck of cards, replace it, and then draw a second card. What is the probability that the first card is an Ace and the second card is a Jack? 5. You draw one card from a deck of cards, remove it, and then draw a second card. What is the probability that the first card is a heart and the second card is a spade? 6. You draw one card from a deck of cards, replace it, and then draw a second card. What is the probability that the first card is a heart and the second card is a Queen?

6 7. You draw one card from a deck of cards, remove it, and then draw a second card. What is the probability that the first card is a spade and the second card is a spade? 8. Suppose Ralph gets a strike when bowling 30% of the time. a. What is the probability that Ralph gets two strikes in a row? b. What is the probability that Ralph gets a turkey (three strikes in a row)? c. When events are independent, their complements are independent as well. Use this result to determine the probability that Ralph gets a strike and then does not get a strike. 9. Suppose a basketball player is an excellent free throw shooter and makes 91% of his free throws (i.e. has a 91% chance of making a single free throw). Assume that free throw shots are independent of one another. Suppose this player gets to shoot two free throws. a. Find the probability that he makes BOTH. b. Find the probability that he makes the first, but misses the second. c. Find the probability that he misses BOTH. 10. A bag contains 25 used golf balls. There are 10 Titlests, 7 Maxflis, 3 Nikes, and 5 Topflight golf balls contained within. a. What is the probability that two Topflight golf balls are chosen in a row (no replacement)? b. What is the probability that a Maxfli ball is chosen first followed by a Titlest (no replacement)?