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Section 7B: Combining Probabilities Section 7B: Combining Probabilities Example. Suppose you roll a fair six-sided die twice. What is the probability that you will get a 2 on the first roll, and then an odd number on the second roll? And Probability: Independent Events Two events are independent if the outcome of one does not affect the probability of the other event. Consider two independent events A and B with individual probabilities P (A) and P (B). The probability that A and B occur together is P (A and B) = P (A) × P (B) This principle can be extended to any number of independent events. For example, the probability of three events A, B, and C all occurring is P (A and B and C) = P (A) × P (B) × P (C) Example. I You flip a coin twice. What is the probability that you will get heads both times? I You flip a coin three times. What is the probability that you will get heads, then tails, then heads again? I You roll a fair six-sided die twice. What is the probability that you will get snake-eyes (two 1’s)? box-cars (two 6’s)? I Being an avid minigolfer, the probability you make a hole-in-one is 0.15. What is the probability that on your next two holes, you don’t make a hole in one? 1 Section 7B: Combining Probabilities Example. There are 10 freshmen and 15 sophomores in a class, and two must be selected for the school senate. If each person is equally likely to be selected, what is the probability that both people selected are sophomores? And Probability: Dependent Events Two events are dependent if the outcome of one event affects the probability of the other event. The probability that dependent events A and B occur together is P (A and B) = P (A) × P (B given A) where P (B given A) means “the probability of event B given the occurrence of event A.” This principle can be extended to any number of dependent events. For examples, the and probability of three dependent events A, B, and C is P (A and B and C) = P (A) × P (B given A) × P (C given A and B) Example. I In the example above, what is the probability that those selected are both freshmen? I 2 Ten names are thrown in a hat for winning a raffle, eight women and two men. Three tickets are drawn out. What is the probability that they will all be women? all men? Section 7B: Combining Probabilities Example. You roll a standard fair six-sided die once. What is the probability that you will get a 4 or an odd value? Either/Or Probability: Non-Overlapping Events Two events are non-overlapping if they cannot occur together. If A and B are non-overlapping events, the probability that either A or B occurs is P (A or B) = P (A) + P (B) This principle can be extended to any number of non-overlapping events. For example, the probability that either event A, event B, or event C occures is P (A or B or C) = P (A) + P (B) + P (C) Example. I You flip a coin three times. What is the probability of getting all heads or all tails? I 3 You draw a card from a standard 52-card deck. What is the probability the card will be a spade or a heart? Section 7B: Combining Probabilities Example. You roll two standard fair six-sided dice. What is the probability that at least one of the dice is a 2 (that is, the first die is a 2 or the second die is a two?) Either/Or Probability: Overlapping Events Two events are overlapping if they can occur together. If A and B are overlapping events, the probability that either A or B occurs is P (A or B) = P (A) + P (B) − P (A and B) Example. Refer to the previous example. What is the probability that at least one die has an odd value? You draw a card from a standard 52-card deck. What is the probability that you get a jack or a spade? 4 Section 7B: Combining Probabilities Example. You draw two cards from a standard 52-card deck. What is the probability that I Both cards are red? I Both cards are 10? Example. You roll two dice. What is the probability that I at least one is a multiple of 3? I they sum to 7 or 11? I their sum is even? I their sum is odd? 5 Section 7B: Combining Probabilities Example. You flip an unfair coin three times. The probability of getting heads is probability of getting all outcomes the same? 2 . 3 What is the Example. A poll is given to a large sampling of students. The poll found that 40% of those sampled like their math class. Additionally, it found that 55% like cheeseburgers. I Assume these events are independent (that is, liking math and liking cheeseburgers are unrelated). What is the probability that someone polled liked both math and cheeseburgers? liked math and disliked cheeseburgers? I Now, assume that these events are overlapping, and the percentage of those who liked both is 15%. What is the probability that that someone polled would like neither math nor cheeseburgers. I Now, assume these events are disjoint. What is the probability that a person polled likes both math and cheeseburgers? Likes neither math nor cheeseburgers? 6