The country of Statland runs a national lottery for its people. In this lottery, an urn contains 42 balls that are individually numbered 1, 2, ..., 42. Each week, 5 balls are drawn from the urn without replacement. Gamblers pay a small sum to predict these five numbers and the jackpot is won by anyone who has chosen all five correctly. A runners-up prize is won by anyone who correctly selected four of the five numbers. You have decided to play the Statland lottery and have chosen your five numbers at random from the 42 available. Give your answer to seven decimals in each part.

What is the probability that you win the jackpot?
What is the probability that you win a runners-up prize?

1.0x10^-6

Please

I want someone who can give me winning numbers

To calculate the probability of winning the jackpot or a runners-up prize in the Statland lottery, we need to determine the number of favorable outcomes (i.e., winning combinations) and the total number of possible outcomes (i.e., all combinations).

1. Probability of winning the jackpot:
To win the jackpot, you need to correctly choose all five numbers out of the 42 available. The number of winning combinations is 1 since there is only one specific combination that corresponds to the drawn balls. The total number of possible combinations is calculated using the binomial coefficient (also known as "n choose k"), which represents the ways to choose k elements from a set of n elements without replacement. In this case, we have 42 balls and need to choose 5. Thus, the total number of combinations is given by C(42, 5).

Therefore, the probability of winning the jackpot is:
P(jackpot) = (Number of winning combinations) / (Total number of combinations)
= 1 / C(42, 5)

2. Probability of winning a runners-up prize:
To win a runners-up prize, you need to correctly choose four of the five numbers drawn from the urn. The number of winning combinations is calculated by choosing 4 out of 5 numbers correctly and choosing 1 number incorrectly. This can be computed as [C(5, 4) * C(37, 1)], where C(a, b) represents the binomial coefficient, as explained earlier. The total number of possible combinations for the runners-up prize is C(42, 5), as all five numbers must be drawn from the 42 available.

Therefore, the probability of winning a runners-up prize is:
P(runners-up) = (Number of winning combinations) / (Total number of combinations)
= [C(5, 4) * C(37, 1)] / C(42, 5)

Now let's calculate the probabilities using the formulas provided:

1. Probability of winning the jackpot:
P(jackpot) = 1 / C(42, 5)
= 1 / (42! / (5! * (42-5)!))
≈ 0.000002087
(Rounded to seven decimal places)

2. Probability of winning a runners-up prize:
P(runners-up) = [C(5, 4) * C(37, 1)] / C(42, 5)
= [5! / (4! * (5-4)!) * 37!] / (42! / (5! * (42-5)!))
≈ 0.000165504
(Rounded to seven decimal places)