Question

In a certain state's lottery, 40 balls numbered 1 through 40 are placed in a machine and 7 of them are drawn at random. If the 7 numbers drawn match the 7 numbers a player has chosen, in any order, she or he win $2,500,000. Alternatively, if 6 of the numbers drawn match 6 of the 7 numbers a player has chosen, in any order, the player wins $40,000, and if 5 of the numbers drawn match 5 of the 7 numbers a player has chosen, in any order, the player wins $10,000.

Required:
a. What is the probability a player who buys one ticket will win the $2,500,000 prize?
b. What is the probability a player who buys one ticket will win the $10,000 prize?

Answer 1

Explanation:

From the given information;

The total number of ways to choose 7 number =
`^(40)C_7`

Number of ways to choose 7 correct numbers =
`^(7)C_7`

∴

The probability P( win $2500000) is;

`= (^(7)C_7)/(^(40)C_7)`

`= ((7!)/(7!(7-7)!) )/((40!)/(7!(40-7)!))`

`= (1 )/((40!)/(7!(40-7)!))`

`= (1 )/((40!)/(7!(33)!))`

`= (1 )/(18643560)`

= 5.36 × 10⁻⁸

The probability P( win $10000) is:

`= (^7C_5 * ^(33) C_2)/(^(40)C_7)`

`= ( (7!)/(5!(7-5)!) * (33!)/(2!(33-2)!) )/( (40!)/(7!(40-7)!))`

`= ( (7!)/(5!(2)!) * (33!)/(2!(31)!) )/( (40!)/(7!(33)!))`

`= ( 21 * 528 )/( 18643560)`

`= ( 11088 )/( 18643560)`

`=(462)/(776815)`

= 5.95 × 10⁻⁴