Probability Chapter 11 1

Independent Events Section

Independent Events  Two events, A and B, are said to be independent if the occurrence of one does not affect the probability of occurrence of the other, and vice versa. Thus, the probability of A and B is defined by 3

Examples A bubble gum machine has 50 cherry-flavored gums, 20 grape-flavored, and 30 licorice-flavored; a second machine has 40 cherry, 50 grape and 10 licorice. A gum is drawn at random from each machine. Find the probability that a. both gums are cherry flavored. b. both gums are licorice flavored. c. the gum from the first machine is cherry flavored, and the one from the second machine is grape flavored. 4

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Examples & Solutions A die is rolled 3 times. Find the probability of Obtaining a. an odd number each time. b. 2 odd numbers first and an even number on the last roll. c. a number less than 5 each time. 6

Examples Assume the spinner has 5 unequal sectors 1/6 red, 1/9 blue, 5/18 green, 2/9 white, and 2/9 yellow. If the spinner is spun twice, find the probability that the spinner will a. land in the white and then in the blue sector. b. land in a sector other than red. c. not land on red both times. d. land on green and red. e. land on the same color both times. 7

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Examples On one of the experimental flights of a space shuttle, the mission was cut short due to a malfunction of a battery aboard the ship. The batteries in the shuttle have a failure rate of 1 in 20. The system of three batteries is designed to operate as long as any one of the batteries functions properly. Find the probability that a. all three batteries fail. b. exactly two fail. c. none of the batteries fail. 9

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Examples & Solutions If a fair coin is tossed 6 times, what is the probability of getting at least 3 heads? 11

Birthday Problem Given a group of people in a room, what is the probability that at least two of the people have the same birthday. Assuming that all birthdays are equally likely, the probability that a second person has a different birthday than the first is 364/ 365, the probability that a third individual has a different birthday then the other two is 363/365, and so on until the nth person’s probability of a different birthday is (365-n+1)/365. The probability that at least two people have the same birthday is one minus the probability no person in the room has the same birthday. The following formula will give the probability that at least two people in the room will have the same birthday with r being the number of people in the room. It is a permutation problem because order is important for no two people to have the same birthday. 12

Bernoulli Trials or Binomial Experiment Bernoulli Experiment must satisfy the following conditions. 1. The same experiment is repeated several times. 2. There are only two possible outcomes, success and failure. 3. The repeated trials are independent, so that the probability of success remains the same for each trial. 13 Formula Binomial Probability n is the number of independent trials of the experiment. r is the number of successes. p is the probability of success. 1 – p is the probability of failure. n – r is number of failures.

Solutions Examples A machine produces widgets at a defective rate of 3%. Find the probability that in a random sample of twelve a. Exactly four are defective. b. less than two are defective. Notice the two parentheses always sum to one and the two exponents always sum to give n. 14

Example A machine produces widgets at a defective rate of 4%. Find the probability that in a random sample of nine two or more are defective. Solution Determine what numbers are acceptable: 2, 3, 4,…,9 Set up the formula similar to solution b on the previous slide for all of the acceptable numbers. This would be time consuming but it would yield the answer. Alternative Solution Determine what numbers are not acceptable: 0 and 1 Set up the formula similar to solution b on the previous slide for all of the non- acceptable numbers and subtract this answer from one. Sometimes it is easier to work the problem that is not asked than the one that is. 15 END