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Ch. 4 - Probability
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 4 - ProbabilityProblem 4.4.39
Chapter 4, Problem 4.4.39

Pick 10 Lottery For the New York Pick 10 lottery, the player first selects 10 numbers from 1 to 80. Then there is an official drawing of 20 numbers from 1 to 80. The prize of \$500,000 is won if the 10 numbers selected by the player are all included in the 20 numbers that are drawn. Find the probability of winning that prize.

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Step 1: Understand the problem. The player selects 10 numbers from a pool of 80, and the official drawing selects 20 numbers from the same pool. To win, all 10 numbers chosen by the player must be among the 20 numbers drawn. This is a probability problem involving combinations.
Step 2: Calculate the total number of ways to choose 10 numbers from the 80 available. This is given by the combination formula: C(n, k) = n! / (k!(n-k)!), where n is the total number of items, and k is the number of items to choose. Here, n = 80 and k = 10. Use the formula to compute C(80, 10).
Step 3: Calculate the number of favorable outcomes. To win, the 10 numbers chosen by the player must be among the 20 numbers drawn. First, calculate the number of ways to choose 10 numbers from the 20 drawn numbers using the combination formula: C(20, 10).
Step 4: Account for the remaining numbers. After selecting the 10 winning numbers from the 20 drawn, the remaining 70 numbers (80 total - 10 chosen by the player) must not include any of the player's numbers. Calculate the number of ways to choose the remaining 10 numbers from these 70 using the combination formula: C(70, 10).
Step 5: Compute the probability. The probability of winning is the ratio of the number of favorable outcomes to the total number of outcomes. This is given by P = C(20, 10) / C(80, 10). Simplify this expression to find the probability of winning the prize.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and arrangements of objects. In the context of the lottery, it helps determine how many ways a player can choose 10 numbers from a set of 80, which is essential for calculating probabilities.

Probability

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In this lottery scenario, it involves calculating the chances of the player's selected numbers being among the drawn numbers, which is crucial for determining the odds of winning.
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Hypergeometric Distribution

The hypergeometric distribution models the probability of k successes in n draws without replacement from a finite population. In this lottery, it applies because the player selects 10 numbers from a total of 80, and we need to find the probability that all selected numbers are included in the 20 drawn numbers.
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