Question

in one lottery, a player wins the jackpot by matching all five distinct numbers drawn in any order from the white balls (1 through 41) and matching the number on the gold ball (1 through 33). if one ticket is purchased, what is the probability of winning the jackpot?

Explanation:

Step1: Calculate number of ways to choose 5 white - ball numbers

The number of combinations of choosing $r$ items from $n$ items is given by the formula $C(n,r)=\frac{n!}{r!(n - r)!}$. Here, $n = 41$ and $r=5$. So, $C(41,5)=\frac{41!}{5!(41 - 5)!}=\frac{41!}{5!×36!}=\frac{41\times40\times39\times38\times37}{5\times4\times3\times2\times1}=749398$.

Step2: Calculate number of ways to choose 1 gold - ball number

There are 33 ways to choose 1 number from 33 for the gold ball.

Step3: Calculate total number of possible lottery combinations

By the fundamental counting principle, the total number of possible lottery combinations is the product of the number of ways to choose white - ball numbers and the number of ways to choose the gold - ball number. So, the total number of combinations is $749398\times33 = 24730134$.

Step4: Calculate the probability of winning

The probability $P$ of winning (since there is only 1 winning combination) is $P=\frac{1}{24730134}$.

Answer:

$\frac{1}{24730134}$