In a​ lottery, the top cash prize was ​$672 ​million, going to three lucky winners. Players pick five different numbers from 1 to 55 and one number from 1 to 49. A player wins a minimum award of $ 100 by correctly matching two numbers drawn from the white balls​ (1 through 55​) and matching the number on the gold ball​ (1 through 49​). What is the probability of winning the minimum​ award in a fraction?

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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!

In a lottery, the top cash prize was $672 million, going to three lucky winners. Players pick five different numbers from 1 to 55 and one number from 1 to 49. A player wins a minimum award of $100 by correctly matching two numbers drawn from the white balls (1 through 55) and matching the number on the gold ball (1 through 49). What is the probability of winning the minimum award in a fraction?

Explore how to calculate the probability of winning a lottery's minimum prize by matching two white balls (out of 55) and one gold ball (out of 49). Learn key concepts in probability and combinatorics, using step-by-step calculations and relevant formulas. Suitable for students in Grades 10-12.

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