In a certain state's lottery, 40 balls numbered 1 through 40 are placed in a machine and eight of them are drawn at random. If the eight numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. In this lottery, the order the numbers are drawn in does not matter.Compute the probability that you win the million-dollar prize if you purchase a single lottery ticket. Write your answer as a reduced fraction.
Question
In a certain state's lottery, 40 balls numbered 1 through 40 are placed in a machine and eight of them are drawn at random. If the eight numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. In this lottery, the order the numbers are drawn in does not matter.Compute the probability that you win the million-dollar prize if you purchase a single lottery ticket. Write your answer as a reduced fraction.
Solution
The probability of winning the lottery can be calculated using the combination formula, which is used to find the number of possible combinations of a given set where order does not matter.
The formula for combinations is: C(n, k) = n! / [k!(n-k)!]
Where:
- n is the total number of items,
- k is the number of items to choose,
- "!" denotes factorial, which means multiplying all positive integers up to that number.
In this case, n = 40 (the total number of balls) and k = 8 (the number of balls drawn).
So, the total number of possible combinations is: C(40, 8) = 40! / [8!(40-8)!]
This equals to 76,904,685 possible combinations.
The probability of winning is the number of successful outcomes (which is 1, since your ticket has to match exactly the 8 numbers drawn) divided by the total number of outcomes.
So, the probability of winning the lottery with one ticket is 1 / 76,904,685.
Therefore, the probability of winning the lottery with a single ticket is 1/76,904,685.
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