To find the probability of winning the minimum award in the lottery, we need to calculate two separate probabilities:

1. Probability of matching exactly two numbers from the five white balls (out of 55).
2. Probability of matching the one gold ball (out of 49).

Step 1: Total number of ways to choose white balls

The total number of ways to choose 5 white balls from 55 is given by the combination formula: $\binom{55}{5} = \frac{55!}{5!(55-5)!}$ Calculating this: $\binom{55}{5} = \frac{55 \times 54 \times 53 \times 52 \times 51}{5 \times 4 \times 3 \times 2 \times 1} = 3,478,761$ So, there are 3,478,761 ways to choose 5 white balls from 55.

Step 2: Number of ways to match exactly two white balls

The number of ways to match exactly 2 white balls is the combination: $\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10$ Now, for the remaining 3 white balls (that the player doesn't match), they must be chosen from the other 50 white balls (since the player didn't match them). The number of ways to choose these is: $\binom{50}{3} = \frac{50 \times 49 \times 48}{3 \times 2 \times 1} = 19,600$ So, the total number of ways to match exactly 2 white balls and miss 3 others is: $10 \times 19,600 = 196,000$

Step 3: Probability of matching the gold ball

Since there is only 1 gold ball to match out of 49, the probability of matching the gold ball is: $\frac{1}{49}$

Step 4: Total number of favorable outcomes

The total number of favorable outcomes is the product of matching 2 white balls and matching the gold ball: $196,000 \times 1 = 196,000$

Step 5: Total number of possible outcomes

The total number of possible outcomes (choosing 5 white balls and 1 gold ball) is: $3,478,761 \times 49 = 170,459,289$

Step 6: Probability of winning the minimum award

The probability of winning the minimum award is the ratio of favorable outcomes to the total number of outcomes: $\frac{196,000}{170,459,289}$

So, the probability of winning the minimum award is: $\boxed{\frac{196,000}{170,459,289}}$

This is the exact probability in fractional form.

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Would you like further explanation or have any additional questions?

Here are 5 related questions to explore further:

1. How do you compute probabilities for matching exactly 3 or 4 white balls?
2. What is the probability of not matching any numbers in this lottery?
3. How does the probability change if the pool of white balls increases to 60?
4. What is the probability of matching only the gold ball and no white balls?
5. How would you calculate the expected value of a lottery ticket in this scenario?

Tip: When calculating combinations, using a scientific calculator or programming tool can help avoid errors in large factorials!