The country of Statland runs a national lottery for its people. In this lottery, an urn contains $38$ balls that are individually numbered $1, 2, …, 38$. 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 $38$ available. Give your answer to seven decimals in each part.
a)
What is the probability that you win the jackpot?
I have reasoned that it is $^{38}C_5=501942$ (combinations).
b)
What is the probability that you win a runners-up prize?
I have reasoned that it is $^{38}C_4=73815$ (combinations).
What are the correct answers and solutions?
Best Answer
a) To compute the probability of winning this lottery you need to divide the number of outcomes where you win by the number of total possible outcomes. There is 1 outcome where you win (where each of the drawn balls in yours) and there are $^{38}C_5 = 501942$ total possible outcome so the probability is $\frac{1}{501942}$.
b) Similarly to win the runner up prize there are $^{5}C_{4} \times {}^{(38-5)}C_{1}$ possible outcomes. This is because 4 of your 5 numbers are drawn and then some other of $38 - 5 = 33$ balls is drawn. There are still $^{38}C_5 = 501942$ outcomes and so the probability is: $$\frac{^{5}C_{4} \times {}^{33}C_{1}}{^{38}C_5}.$$