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 42) and matching the number on the gold ball (1 through 32). if one ticket is purchased, what is the probability of winning the jackpot?the probability of winning the jackpot with one ticket is \\(\square\\).(type an integer or a simplified fraction.)

Explanation:

Step1: Calculate combinations for white balls

We need to find the number of ways to choose 5 distinct numbers from 42, which is given by the combination formula \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 42 \) and \( k=5 \).
\[

$$\begin{align*} C(42,5)&=\frac{42!}{5!(42 - 5)!}\\ &=\frac{42!}{5!×37!}\\ &=\frac{42\times41\times40\times39\times38}{5\times4\times3\times2\times1}\\ &=\frac{42\times41\times40\times39\times38}{120}\\ &=850668 \end{align*}$$

\]

Step2: Calculate number of possibilities for gold ball

There are 32 possible numbers for the gold ball, so there are 32 ways to choose the gold ball number.

Step3: Calculate total number of possible outcomes

The total number of possible lottery outcomes is the product of the number of ways to choose the white balls and the number of ways to choose the gold ball. So total outcomes \(=C(42,5)\times32=850668\times32 = 27221376\)

Step4: Calculate probability

The probability of winning the jackpot with one ticket is the reciprocal of the total number of possible outcomes since there is only 1 winning combination. So probability \(=\frac{1}{27221376}\)

Answer:

\(\frac{1}{27221376}\)