Question

There are 15 tennis balls in a box, of which 9 have not previously been used. three of the balls are randomly chosen, played with, and then returned to the box. later, another 3 balls are randomly chosen from the box. find the probability that none of these balls has ever been used

Answer 1

Total tennis balls = 15 balls
Previously not used balls = 9 balls
Previously used balls = 15 - 9 = 6 balls

Probability of getting first 3 balls that are not used previously =
`(9)/(15)` ×
`(8)/(14)` ×
`(7)/(13)`

Now remaining total balls = 15 - 3 = 12
Balls that are not previously used = 9 - 3 = 6

Probability of getting next 3 balls that are not used previously =
`(6)/(12)` ×
`(5)/(11)` ×
`(4)/(10)`

Probability =
`(9)/(15)` ×
`(8)/(14)` ×
`(7)/(13)` ×
`(6)/(12)` ×
`(5)/(11)` ×
`(4)/(10)`

= 0.089

Answer 2

For the first three balls that are chosen, the probability would be found by the following expression: 9/15*8/14*7/13 = 12/65  The numerator decreases since the number of new balls is decreasing with each one that is removed. The denominator decreases since the total number of tennis balls in the box is being decreased. For the second time that balls are chosen, the following expression is used: 6/15*5/14*4/13 = 10/273 The numerator starts with 6 since there are now fewer balls that are unused due to the ones previously have been used. These two fractions are then multiplied to find the total probability that none of the 6 balls were used. 12/65*10/273 = 8/1183