712 CHAPTER 11 Probability The probability of an event often depends on other conditions related to the event. For example, the probability that it rains on a given day depends on many conditions such as humidity level, atmospheric pressure, wind speed, and temperature. The probability that a new electric car model sells well may depend on the price of the car, the overall state of the economy, the advertising budget for the car, and the sales of competing cars. Probability problems such as these, where the outcomes of related events affect the outcome of a given event, are called conditional probability problems. SECTION 11.6 Conditional Probability LEARNING GOAL Upon completion of this section, you will be able to: 7 Solve conditional probability problems. Why This Is Important In addition to applications in weather forecasting and business modeling, conditional probability has many applications in science, engineering, and finance. In Section 11.5, we learned that when two events are dependent, the occurrence of the first event, A, affected the probability of the second event, B, occurring. When we calculated P A( and B), when determining the probability of event B, we assumed that event A occurred. That is, we calculated the probability of event B, given event A. The probability of event B, given event A, is called conditional probability. The definition of conditional probability follows. 10-peso coin, and three 20-peso coins. He selects two coins without replacement. Assuming that each coin is equally likely to be selected, determine the probability that Martin selects at least one 1-peso coin. 8 15 101. A Fair Game? Two playing cards are dealt to you from a standard 52-card deck. If either card is a diamond or if both are diamonds, you win; otherwise, you lose. Determine whether this game favors you, is fair, or favors the dealer. Explain your answer. Favors dealer. The probability of at least one diamond is 0.44, ≈ which is less than 0.5. 102. Face Card Probability You have three cards: an ace, a king, and a queen. A friend shuffles the cards, randomly selects two of them, and discards the third. You ask your friend to show you a face card, and she turns over the king. What is the probability that she also has the queen? 1 2 Recreational Mathematics A Different Die For Exercises 103–106, consider a six-sided die that has 1 dot on one side, 2 dots on two sides, and 3 dots on three sides. If the die is rolled twice, determine the probability of rolling 103. two 2’s. 1 9 104. two 3’s. 1 4 If the die is rolled only once, determine the probability of rolling 105. an even number or a number less than 3. 1 2 106. an odd number or a number greater than 1. 1 107. The Birthday Problem Read the Did You Know? on page 706. Use this information to calculate the probability that at least two people in your class have the same birthday. Poll your classmates to determine whether two or more of them have the same birthday. Answers will vary Research Activity 108. Girolamo Cardano Girolamo Cardano (1501–1576) wrote Liber de Ludo Aleae, which is considered to be the first book on probability. Cardano had a number of different vocations. Do research and write a paper on the life and accomplishments of Girolamo Cardano. Naypong Studio/ Shutterstock
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