Probability Chapter 11 1

Sample Spaces and Probability Section 11.1 2

Probabilities The probability of a given event is a mathematical estimate of the likelihood that this event will occur. Activities such as tossing a coin, drawing a card from a deck, or rolling a pair of dice are called experiments. The set U of all possible outcomes for an experiment is called the sample space. Each of the subsets of U that can occur is called an event. 3

Theoretical Probability Suppose an experiment has n(U) possible outcomes, all equally likely. Suppose further that the event E occurs in n(E) of these outcomes. Then, the probability of event E is given by 4

Probability of an Event Not Occurring The probability P(E’ ) of an event not occurring is Thus, the probability P(E) of at least one is 5

Examples On a single toss of a die, what is the probability of obtaining the following? The number 5. An even number. A number greater than 4. 6

Examples A single ball is taken at random from an urn containing 10 balls numbered 1 through 10. What is the probability of obtaining the following? Ball number 8. A ball different from 5. A ball numbered 12. 7

Examples Assume that a single card is drawn from a well-shuffled deck of 52 cards. Find the probability of the following: One of the face cards is drawn. A face card or a spade is drawn. 8

Empirical Probability The empirical probability of event E is 9

Examples Write the answers as a fraction in reduced form. A spinner has 4 equal sectors colored yellow, blue, green, and red. If in 100 spins we get 25 blue, 28 red, 24 green, and 23 yellow outcomes, find the empirical probability of getting blue P(B). getting red P(R). getting green P(G). getting yellow P(Y). What are the theoretical probabilities of P(B), P(R), P(G), and P(Y)? Which outcome has the same empirical and theoretical probability? 10

Examples 11

Examples Write the answers as a fraction in reduced form. 35. Suppose 2 dice are rolled 50 times and that the results are shown in the bar graph on P713. Find the empirical probability of P(11), the sum is 11. P(7), the sum is 7. P(O), the sum is odd. P(~O), the sum is not odd. Which three outcomes have the same empirical probability? Do these outcome have the same empirical and theoretical probability? 12

Examples 13

Examples A machine produces widgets at a defective rate of 2%. A sample of 5 widgets is taken. What is the probability that 2 of them are defective? 14

An urn contains 9 white, 7 red, 5 green, and 8 blue chips An urn contains 9 white, 7 red, 5 green, and 8 blue chips. If four chips are picked at random, what is the probability of two red chips? First determine how many colors the question indicated are in the urn. The answer is not four colors. The question indicates there are two colors: red and not red. n(R) = 7 ; n(R’) = 22 ; n(U) = 29 Next we determine order is not important and therefore makes this a combination problem. 7 nCr 2  22 nCr 2 29 nCr 4 = 77 377 = 0.2042 P(E) = If six chips are chosen at random, what is the probability of getting 1 green and 2 white chips? There are now three colors in the urn: green, white and other. Again order is not important. 15

n(G) = 5 ; n(W) = 9 ; n(other) = 15; and n(U) = 29 5 nCr 1  9 nCr 2  15 nCr 3 29 nCr 6 P(E) = = 5 29 = 0.1724 END 16