QuestionJuly 14, 2025

In one lottery game contestants pick five numbers from 1 through 31 and have to match all five for the big prize (in any order). You'll get twice your money back if you match three out of five numbers. If you buy five tickets, what's the probability of matching three out of five numbers? If you buy five tickets, the probability of matching three out of five numbers is square (Enter your answer as a fraction in lowest terms.)

In one lottery game contestants pick five numbers from 1 through 31 and have to match all five for the big prize (in any order). You'll get twice your money back if you match three out of five numbers. If you buy five tickets, what's the probability of matching three out of five numbers? If you buy five tickets, the probability of matching three out of five numbers is square (Enter your answer as a fraction in lowest terms.)
In one lottery game contestants pick five numbers from 1 through 31 and have to match all five for the big prize (in any order).
You'll get twice your money back if you match three out of five numbers. If you buy five tickets, what's the probability of matching three out of five numbers?
If you buy five tickets, the probability of matching three out of five numbers is square 
(Enter your answer as a fraction in lowest terms.)

Solution
4.1(306 votes)

Answer

\frac{1625}{33982} Explanation 1. Calculate Total Combinations The total number of ways to choose 5 numbers from 31 is given by the combination formula: **C(n, r) = \frac{n!}{r!(n-r)!}**. Here, n = 31 and r = 5. So, C(31, 5) = \frac{31!}{5!(31-5)!} = 169,911. 2. Calculate Favorable Combinations for Matching 3 Numbers Choose 3 correct numbers out of 5: C(5, 3) = 10. Choose 2 incorrect numbers from the remaining 26: C(26, 2) = 325. Total favorable combinations = 10 \times 325 = 3,250. 3. Calculate Probability for One Ticket Probability of matching exactly 3 numbers with one ticket is \frac{3,250}{169,911}. 4. Calculate Probability for Five Tickets Probability of not matching 3 numbers in one ticket is 1 - \frac{3,250}{169,911}. For five tickets, probability of not matching 3 numbers in any ticket is (1 - \frac{3,250}{169,911})^5. Therefore, probability of matching 3 numbers in at least one ticket is 1 - (1 - \frac{3,250}{169,911})^5. 5. Simplify the Expression Calculate the above expression to get the final probability.

Explanation

1. Calculate Total Combinations<br /> The total number of ways to choose 5 numbers from 31 is given by the combination formula: **$C(n, r) = \frac{n!}{r!(n-r)!}$**. Here, $n = 31$ and $r = 5$. So, $C(31, 5) = \frac{31!}{5!(31-5)!} = 169,911$.<br /><br />2. Calculate Favorable Combinations for Matching 3 Numbers<br /> Choose 3 correct numbers out of 5: $C(5, 3) = 10$. Choose 2 incorrect numbers from the remaining 26: $C(26, 2) = 325$. Total favorable combinations = $10 \times 325 = 3,250$.<br /><br />3. Calculate Probability for One Ticket<br /> Probability of matching exactly 3 numbers with one ticket is $\frac{3,250}{169,911}$.<br /><br />4. Calculate Probability for Five Tickets<br /> Probability of not matching 3 numbers in one ticket is $1 - \frac{3,250}{169,911}$. For five tickets, probability of not matching 3 numbers in any ticket is $(1 - \frac{3,250}{169,911})^5$. Therefore, probability of matching 3 numbers in at least one ticket is $1 - (1 - \frac{3,250}{169,911})^5$.<br /><br />5. Simplify the Expression<br /> Calculate the above expression to get the final probability.
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