Questions: In a lottery, the top cash prize was 647 million, going to three lucky winners. Players pick five different numbers from 1 to 54 and one number from 1 to 47. A player wins a minimum award of 600 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 (Type an integer or a simplified fraction.)

Solution Steps

To win the minimum award, a player must match exactly 3 out of 5 white balls picked. The number of ways to choose 3 winning white balls from the 5 picked by the player is \(\binom{5}{3} = 10\). The number of ways to choose the remaining 2 white balls from the 49 non-winning white balls is \(\binom{49}{2} = 1176\). The total number of ways to choose 5 white balls from 54 is \(\binom{54}{5} = 3162510\). The probability of matching exactly 3 white balls is then \(\frac{\binom{5}{3} \cdot \binom{49}{2}}{\binom{54}{5}} = 0.00372\).

Since there is only one gold ball to be picked correctly out of 47, the probability is \(\frac{1}{47} = 0.0213\).

The total probability of winning the minimum award by matching exactly 3 white balls and the gold ball is: Total probability = 0.00372 \times 0.0213 = 7.91184\times 10^{-5}

Final Answer: