Question

A certain lottery game asks a person to choose five numbers between 1 and 50 (nonrepeating) and then one "powerball" number between 1 and 30. In order to win the jackpot, all 6 numbers must match.

(a) What is the probability of winning the jackpot?

(b) To win the second grand prize, the person must match only 3 of the 5 numbers and cannot match the powerball number. What is the probability of winning the second prize?

Answer 1

Number of ways in which r items can be selected from n, nCr = n!/(r! x (n-r)!)

a) Number of ways in which 5 winning numbers can be chosen from 50 = 50C5

= 50!/(5! x 45!)

= 2,118,760

Number of ways in which 1 powerball can be selected = 30

P(winning the jackpot) = 1/(2,118,760 x 30)

= 1/63,562,800

= 1.573x10^-8

b) P(winning second grand prize) = P(3 numbers match the any 3 of the 5 winning numbers x 2 numbers match the non winning 44 numbers) x P(not getting powerball number)

= (5C3 x 44C2 / 50C5) x (29/30)

= 0.00432