In 1990, California switched from a 6/49 lottery to a 6/53 lottery. Later, the state switched again to a 6/51 lottery. Find the probability of winning first prize in a 6/49 lottery.

To find the probability of winning the first prize in a 6/49 lottery, we need to calculate the number of possible combinations of 6 numbers that can be selected from a total of 49 numbers. The probability is given by the inverse of this number of combinations, as there is only one winning combination.

### Step 1: Calculate the number of combinations
The number of combinations of 6 numbers out of 49 is given by the combination formula:

\[
\text{Number of combinations} = \binom{49}{6} = \frac{49!}{6!(49-6)!} = \frac{49!}{6! \times 43!}
\]

### Step 2: Simplify the expression
Now, let's calculate this step by step:

\[
\binom{49}{6} = \frac{49 \times 48 \times 47 \times 46 \times 45 \times 44}{6 \times 5 \times 4 \times 3 \times 2 \times 1}
\]

\[
\binom{49}{6} = \frac{10068347520}{720} = 13,983,816
\]

### Step 3: Calculate the probability
The probability of winning is the reciprocal of the number of combinations:

\[
P(\text{winning}) = \frac{1}{\binom{49}{6}} = \frac{1}{13,983,816}
\]

### Final Answer
The probability of winning the first prize in a 6/49 lottery is:

\[
P(\text{winning}) = \frac{1}{13,983,816}
\]

Would you like more details or have any questions?

Here are 5 related questions you might be interested in:
1. What is the probability of winning a 6/53 lottery?
2. How does the probability change when the number of selections is reduced to 5 out of 49?
3. What are the odds of winning any prize in a 6/49 lottery if additional lower-tier prizes are available?
4. How does increasing the total number of numbers (e.g., 6/60) affect the probability of winning?
5. Can you compare the probabilities of winning between different lottery systems?

**Tip:** Remember that as the number of possible combinations increases (due to more numbers or more choices), the probability of winning decreases.

Discover how to calculate the probability of winning the first prize in a 6/49 lottery. Learn about combinatorics and the binomial coefficient formula used to determine the number of combinations. Suitable for Grades 10-12.

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