148 CHAPTER 3 Probability Independent and Dependent Events In some experiments, one event does not affect the probability of another. For instance, when you roll a die and toss a coin, the outcome of the roll of the die does not affect the probability of the coin landing heads up. These two events are independent. The question of the independence of two or more events is important to researchers in fields such as marketing, medicine, and psychology. You can use conditional probabilities to determine whether events are independent. Two events are independent when the occurrence of one of the events does not affect the probability of the occurrence of the other event. Two events A and B are independent when P1B0 A2 = P1B2 Occurrence of A does not affect probability of B or when P1A0 B2 = P1A2. Occurrence of B does not affect probability of A Events that are not independent are dependent. DEFINITION To determine whether A and B are independent, first calculate P1B2, the probability of event B. Then calculate P1B0 A2, the probability of B, given A. If the values are equal, then the events are independent. If P1B2 ≠ P1B0 A2, then A and B are dependent events. Classifying Events as Independent or Dependent Determine whether the events are independent or dependent. 1. Selecting a king 1A2 from a standard deck of 52 playing cards, not replacing it, and then selecting a queen 1B2 from the deck 2. Tossing a coin and getting a head 1A2, and then rolling a six-sided die and obtaining a 6 1B2 3. Driving over 85 miles per hour 1A2, and then getting in a car accident 1B2 SOLUTION 1. P1B2 = 4 52 and P1B0 A2 = 4 51. The occurrence of A changes the probability of the occurrence of B, so the events are dependent. 2. P1B2 = 1 6 and P1B0 A2 = 1 6. The occurrence of A does not change the probability of the occurrence of B, so the events are independent. 3. Driving over 85 miles per hour increases the chances of getting in an accident, so these events are dependent. TRY IT YOURSELF 2 Determine whether the events are independent or dependent. 1. Smoking a pack of cigarettes per day 1A2 and developing emphysema, a chronic lung disease 1B2 2. Tossing a coin and getting a head 1A2, and then tossing the coin again and getting a tail 1B2 Answer: Page A38 EXAMPLE 2 Picturing the World Truman Collins, a probability and statistics enthusiast, wrote a program that finds the probability of landing on each square of a Monopoly® board during a game. Collins explored various scenarios, including the effects of the Chance and Community Chest cards and the various ways of landing in or getting out of jail. Interestingly, Collins discovered that the length of each jail term affects the probabilities. (Note that the probabilities are rounded to more than three decimal places so that it is easier to see how going to jail affects the probabilities.) Monopoly square Probability given short jail term Probability given long jail term Go 0.0310 0.0291 Chance 0.0087 0.0082 In Jail 0.0395 0.0946 Free Parking 0.0288 0.0283 Park Place 0.0219 0.0206 B&O RR 0.0307 0.0289 Water Works 0.0281 0.0265 Why do the probabilities depend on how long you stay in jail?
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