In this lottery, the player wins the jackpot by matching:

• Five distinct numbers drawn from the 41 white balls, and
• One number drawn from the 34 gold balls.

To calculate the probability of winning, we need to find the total number of possible outcomes (the sample space) and the number of favorable outcomes (the winning outcome).

Step 1: Total number of possible outcomes

• White balls: The player needs to choose 5 distinct numbers from a set of 41 white balls. Since the order of selection doesn't matter, this is a combination problem. The number of ways to choose 5 numbers from 41 is given by the combination formula: $\binom{41}{5} = \frac{41!}{5!(41 - 5)!}$

• Gold ball: There are 34 possible outcomes for the gold ball (the number drawn from 1 to 34).

Thus, the total number of possible outcomes is: $\text{Total outcomes} = \binom{41}{5} \times 34$

Step 2: Number of favorable outcomes

There is only one favorable outcome, where the player matches all 5 white balls and the correct gold ball.

Step 3: Probability of winning

The probability $P$ of winning is the ratio of favorable outcomes to total outcomes: $P = \frac{1}{\binom{41}{5} \times 34}$

Now, let's calculate the values:

• First, calculate $\binom{41}{5}$: $\binom{41}{5} = \frac{41 \times 40 \times 39 \times 38 \times 37}{5 \times 4 \times 3 \times 2 \times 1} = 4,496,388$

• Now, calculate the total number of outcomes: $\text{Total outcomes} = 4,496,388 \times 34 = 152,780,232$

• Therefore, the probability of winning the jackpot is: $P = \frac{1}{152,780,232}$

Final Answer:

The probability of winning the jackpot is approximately: $P \approx 6.55 \times 10^{-9}$ which is a very small probability.

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Would you like more details or have any questions on this? Here are a few related topics you might find interesting:

1. How to calculate combinations and permutations in probability.
2. Understanding the odds in different types of lotteries.
3. How to calculate the expected value of a lottery ticket.
4. Comparing the probabilities of winning in different lottery formats.
5. Exploring the role of combinatorics in probability problems.

Tip: When dealing with lottery problems, combinations are often used to determine how many ways something can happen without regard to order. This contrasts with permutations, where order matters.