Question

In the lottery game called 5/21, a player picks five numbers from 1 to 21. Enter all answers as a decimal (not a percent), rounded to five decimal places.

a) If all five of the numbers match the ones that are drawn, the player wins first prize. What is the probability of winning this prize?

b) If four of the five numbers match the ones that are drawn, but the other number doesn't match, the player wins second prize. What is the probability of winning this prize?

c) If three of the five numbers match the ones that are drawn, but the other two numbers don't match, the player wins third prize. What is the probability of winning this prize?

Answer 1

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

Total number of ways in which 5 numbers can be selected = 21C5 = 20,349

a) Number of ways in which 5 numbers can match the ones that are drawn = 5C5 = 1

P(the player wins first prize) = 1/20,349

= 0.00005

b) Number of ways in which 4 numbers can match the winning 5 and 1 number can match the non winning 16 = 5C4 x 16C1

= 5 x 16

= 80

P(the player wins second prize) = 80/20,349

= 0.00393

c) Number of ways in which 3 numbers can match the winning 5 and 2 numbers can match the non winning 16 = 5C3 x 16C2

= 10 x 120

= 1200

P(the player wins third prize) = 1200/20,349

= 0.05897