Question

A cricket club has 16 members, of whom only 5 can bowl. What is the probability that in a team of 11 members at least 3 bowlers are selected?

Answer 1

Number of members in the cricket club = 16

Number of bowlers = 5

Number of batters = 16 – 5 = 11

The probability that a team of 11 members consisting of atleast 3 bowlers = (Probability of selecting 3 bowlers and 8 batters) + ( Probability of selecting 4 bowlers and 7 batters) + (Probability of selecting 5 bowlers and 6 batters)

The probability that a team of 11 members consisting of atleast 3 bowlers

= `(""^5"C"_3 xx ""^11"C"_8)/(""^16"C"_11) + (""^5"C"_4 xx ""^11"C"_7)/(""^16"C"_11) + (""^5"C"_5 xx ""^11"C"_6)/(""^16"C"_11)`

Selection procedure:

First out of total 16 members selecting 11 members in 16C₁₁ ways.

Selection of 11 members consisting minimum of 3 bowlers.

∴ Selection of 11 members as follows

(1) 3 bowlers from 5 bowlers and 8 batters from 11 batters.

(2) 4 bowlers from 5 bowlers and 7 batters from 11 batters.

(3) 5 bowlers from 5 bowlers and 6 batters from 11 batters.

= `(""^5"C"_2 xx ""^11"C"_3 + ""^5"C"_1 xx ""^11"C"_4 + 1 xx ""^11"C"_5)/(""^16"C"_5)`

= `((5 xx 4)/(1 xx 2) xx (11 xx 10 xx 9)/(1 xx 2 xx 3) + 5 xx (11 xx 10 xx 9 xx 8)/(1 xx 2 xx 3 xx 4) + (11 xx 10 xx 9 xx 8 xx 7)/(1 xx 2 xx 3 xx 4 xx 5))/((16 xx 5 xx 14 xx 13 xx 12)/(1 xx 2 xx 3 xx 4 xx 5))`

= `((5 xx 2) xx (11 xx 5 xx 3) + 5 xx (11 xx 10 xx 3) + 11 xx 3 xx 2 xx 7)/(4 xx 3 xx 14 xx 13 xx 2)`

= `(10 xx 165 + 5 xx 330 + 33 xx 14)/(24 xx 14 xx 13)`

= `(1650 + 1650 + 462)/4368`

= `3762/4368`

= `627/728`