Question

In a closet there are 10 pairs of shoes. If six shoes are selected at random, what is the probability of (a) no complete pairs; (b) exactly one complete pair; (c) exactly two complete pairs; (d) exactly three complete p airs?

Answer 1

Here there are 10 pairs of shoes

now we selected 6 shoes at random

Here we have to find the probability of

(a) No complete pairs so that means first for first shoe there will be 20 shoes to chose then for the second one there are 18 and then for third one 16 and so on

so,

Pr(No complete pairs) = (20 * 18 * 16 * 14 * 12 * 10)/ (20 * 19 * 18 * 17 * 16 * 15) = =  ¹⁰C₆ 2⁶ / ²⁰C₆= 0.3467

(b) Nowthere is exactly one pair. so there are 10 ways to chose a pair and for remaining 9 pairs they we should chose 4 shoes like last question

Pr(one complete pair) = 10 * (18 * 16 * 14 * 12)/(20 * 19 * 18 * 17 * 16 * 15) = 0.5201

(c) Exactly two complete pair , first we chose two pairs which will be ¹⁰C₂ and then remaining 8 pairs and we have too chose 2 shoes out of them

Pr(two completer pair) = ¹⁰C₂ * (16 * 14 )/(20 * 19 * 18 * 17 * 16 * 15) = = 0.1300

(d)

Exactly three complete pair , first we chosethree pairs which can be ¹⁰C₃

Pr(two completer pair) = ¹⁰C₃ /(20 * 19 * 18 * 17 * 16 * 15) = = 0.0031