Question

in a lottery, the top - cash prize was $648 million, going to three lucky winners. players pick four different numbers from 1 to 55 and one number from 1 to 46. a player wins a minimum award of $350 by correctly matching two numbers drawn from the white balls (1 through 55) and matching the number on the gold ball (1 through 46). 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 - balls

We use the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 55\) and \(r=4\). The number of ways to choose 4 white - balls out of 55 is \(C(55,4)=\frac{55!}{4!(55 - 4)!}=\frac{55\times54\times53\times52}{4\times3\times2\times1}=341055\). The number of ways to choose 2 winning white - balls out of 4 winning white - balls is \(C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4\times3}{2\times1}=6\). The number of ways to choose \(4 - 2=2\) non - winning white - balls out of \(55 - 4 = 51\) non - winning white - balls is \(C(51,2)=\frac{51!}{2!(51 - 2)!}=\frac{51\times50}{2\times1}=1275\). The number of ways to choose the white - balls as required is \(C(4,2)\times C(51,2)=6\times1275 = 7650\).

Step2: Calculate number of ways to choose the gold ball

There is 1 winning gold ball out of 40. The number of ways to choose the gold ball correctly is \(C(1,1) = 1\), and the total number of ways to choose a gold ball is \(C(40,1)=40\).

Step3: Calculate total number of ways to play

The total number of ways to choose 4 white - balls out of 55 and 1 gold ball out of 40 is \(C(55,4)\times C(40,1)=341055\times40 = 13642200\).

Step4: Calculate the probability

The probability \(P\) of winning the minimum award is the product of the number of favorable ways to choose white - balls and gold ball divided by the total number of ways to play. \(P=\frac{C(4,2)\times C(51,2)\times C(1,1)}{C(55,4)\times C(40,1)}=\frac{7650\times1}{13642200}=\frac{1}{1783.3}\approx\frac{1}{1783}\).

Answer:

\(\frac{1}{1783}\)