This is part of an exam question from 2014 In the nineteen n

This is part of an exam question from 2014.) In the nineteen nineties, a well-known

American toy company produced a range of talking dolls each of which said four distinct phrases.

Each doll’s phrases were randomly chosen from a list of 270 phrases. This was intended to give the

impression that the dolls were individual.

(a)

What is the exact probability that two such dolls selected at random said the same four phrases

as each other? (Hint: It does not matter what four phrases the first doll says.)

One of the 270 phrases was considered by the public to be inappropriate. The company announced

that customers could return any dolls which said the inappropriate phrase.

(b)

What is the probability that a doll selected at random did not say this phrase?

As order doesn\'t matter here, there are

270C4 = 216546345 ways

to choose 4 phrases.

There is only 1 (one) way for them to say the same.

Thus, the probability that 2 dolls say the same phrase is

P(say the same) = 1/216526345 [ANSWER]

*********************************

Note that

P(did not say) = 1 - P(said)

To get the probability that the doll said the phrase, we fix one of the phrases to be the forbidden phrase.

Thus, there are 3 more phrases to choose, out of 269.

Thus, there are 269C3 = 3208094 ways to do this.

Thus,

P(said) = 3208094/216526345 = 2/135

Thus,

P(did not say) = 133/135 [ANSWER]