SECTION 3.4 Additional Topics in Probability and Counting 173 Finding Probabilities Find the probability of being dealt 5 diamonds from a standard deck of 52 playing cards. SOLUTION In a standard deck of playing cards, 13 cards are diamonds. Note that it does not matter in what order the cards are selected. The possible number of ways of choosing 5 diamonds out of 13 is 13C5. The number of possible five-card hands is 52C5. So, the probability of being dealt five diamonds is P15 diamonds2 = 13 C5 52C5 = 1287 2,598,960 ≈ 0.0005. TRY IT YOURSELF 7 Find the probability of being dealt five diamonds from a standard deck of playing cards that also includes two jokers. In this case, the joker is considered to be a wild card that can be used to represent any card in the deck. Answer: Page A38 Finding Probabilities A food manufacturer is analyzing a sample of 400 corn kernels for the presence of a toxin. In this sample, three kernels have dangerously high levels of the toxin. Four kernels are randomly selected from the sample. What is the probability that exactly one kernel contains a dangerously high level of the toxin? SOLUTION Note that it does not matter in what order the kernels are selected. The possible number of ways of choosing one toxic kernel out of three toxic kernels is 3C1. The possible number of ways of choosing 3 nontoxic kernels from 397 nontoxic kernels is 397C3. So, using the Fundamental Counting Principle, the number of ways of choosing one toxic kernel and three nontoxic kernels is 3C1 # 397C3 = 3# 10,349,790 = 31,049,370. The number of possible ways of choosing 4 kernels from 400 kernels is 400C4 = 1,050,739,900. So, the probability of selecting exactly one toxic kernel is P11 toxic kernel2 = 3 C1 # 397C3 400C4 = 31,049,370 1,050,739,900 ≈ 0.030. TRY IT YOURSELF 8 A jury consists of five men and seven women. Three jury members are selected at random for an interview. Find the probability that all three are men. Answer: Page A38 EXAMPLE 7 Picturing the World One of the largest lottery jackpots ever, $1.6 billion, was won in the Powerball lottery. When the jackpot was won, five different numbers were chosen from 1 to 69 and one number, the Powerball, was chosen from 1 to 26.The winning numbers are shown below. Powerball In 2015, the lottery changed its rules. Previously, a player would choose five different numbers from 1 to 59 and one number from 1 to 35. A player wins the jackpot by matching all six winning numbers in a drawing. You purchase one ticket in the Powerball lottery. Find the probability of winning the jackpot using the old rules and the new rules. Which set of rules provides you with a better chance of winning the jackpot? How likely is your chance of winning? EXAMPLE 8
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