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Question 7, 11.5.9
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In a lottery, the top cash prize was $\$ 653$ million, going to three lucky winners. Players pick four different numbers from 1 to 54 and one number from 1 to 47 .

A player wins a minimum award of $\$ 300$ by correctly matching three numbers drawn from the white balls ( 1 through 54 ) and matching the number on the gold ball (1 through 47). What is the probability of winning the minimum award?

The probability of winning the minimum award is $\square$.
(Type an integer or a simplified fraction.)

Step 1 ：In a lottery, the top cash prize was $653 million, going to three lucky winners. Players pick four different numbers from 1 to 54 and one number from 1 to 47.

Step 2 ：A player wins a minimum award of $300 by correctly matching three numbers drawn from the white balls (1 through 54) and matching the number on the gold ball (1 through 47). We are asked to find the probability of winning the minimum award.

Step 3 ：The total number of ways to draw four numbers from 54 is \( \binom{54}{4} = 316251 \).

Step 4 ：The total number of ways to draw one number from 47 is 47.

Step 5 ：So, the total number of outcomes is \( 316251 \times 47 = 14863797 \).

Step 6 ：The number of successful outcomes is \( \binom{4}{3} \times 1 = 4 \).

Step 7 ：Therefore, the probability of a successful outcome is \( \frac{4}{14863797} = 2.691102414813658 \times 10^{-7} \).

Step 8 ：So, the probability of winning the minimum award is \( \boxed{2.691102414813658 \times 10^{-7}} \).