Question

in a lottery, the top - cash prize was $621 million, going to three lucky winners. players pick four different numbers from 1 to 56 and one number from 1 to 45. a player wins a minimum award of $600 by correctly matching three numbers drawn from the white balls (1 through 56) and matching the number on the gold ball (1 through 45). what is the probability of winning the minimum award? the probability of winning the minimum award is . (type an integer or a simplified fraction.)

Explanation:

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

We use the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 56$ and $r=4$. The number of ways to choose 4 numbers from 56 is $C(56,4)=\frac{56!}{4!(56 - 4)!}=\frac{56\times55\times54\times53}{4\times3\times2\times1}=327630$. The number of ways to choose 3 correct white - ball numbers out of 4 winning white - ball numbers is $C(4,3)=\frac{4!}{3!(4 - 3)!}=4$. The number of ways to choose 1 incorrect white - ball number out of the remaining $56 - 4 = 52$ white balls is $C(52,1)=\frac{52!}{1!(52 - 1)!}=52$.

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

The number of ways to choose 1 number from 45 for the gold ball is $C(45,1)=\frac{45!}{1!(45 - 1)!}=45$, and there is only 1 winning gold - ball number.

Step3: Calculate the total number of favorable outcomes

The number of favorable outcomes for matching 3 white - ball numbers and 1 gold - ball number is $C(4,3)\times C(52,1)\times1=4\times52\times1 = 208$.

Step4: Calculate the total number of possible outcomes

The total number of ways to choose 4 white - ball numbers and 1 gold - ball number is $C(56,4)\times C(45,1)=327630\times45 = 14743350$.

Step5: Calculate the probability

The probability $P$ of winning the minimum award is the number of favorable outcomes divided by the total number of possible outcomes. So $P=\frac{208}{14743350}=\frac{104}{7371675}$.

Answer:

$\frac{104}{7371675}$