Shankar was supposed to come up with the slogan for the STOR department after fa

Question

Shankar was supposed to come up with the slogan for the STOR department after fall break. The slogan has to consist of 4 words . Unfortunately Shankar came across a new anime ( One Punch Man ) over the break and forgot to think up a nice slogan. Shankar knows the pressure is on him as the whole world watches. So he decides on the following algorithm:

1. He creates a collection of words and symbols he likes relevant to the department. The collection consists of the following: “probability”, “is”, “sky”, “limit”, “roof ”, “the”, “!”, “indepen- dence”, “life”, “love”

2. Assume Shankar’s preference for the above four words are given by

P ( probability ) = P ( love ) = P ( independence ) = 1/4

P ( lif e ) = P ( sky ) = P ( limit ) = 1/24

P (!) = P ( roof ) = 1 32 P ( is ) = P ( the ) = 3/64

3. Now Shankar conducts 4 independent trials where each trial is picking one the above collection of words and symbols with the preferences above. Note: our assumption is that the trials are independent that is even if the first 2 trials are “roof”, “roof”, this does not effect the last two trials.

Let W ords denote the random collection of words and symbols (written down in no particular order) that Shankar picks at the end of the 4 trials. Thus for example suppose

Words = Roof, Roof, Roof, !

This means on 3 of the 4 trials, Shankar picked Roof (not necessarily the first 3), and on the remaining trial Shankar picked !.

(a) Compute the probability that Words = Probability, is, lif e, !

(b) Compute the probability that Words = Probability, ! , ! , !

(c) Compute the probability that Words = Independence, is, the, limit

Explanation / Answer

Solution

(a) Probability= 4!*((1/4)*(3/64)*(1/24)*(1/32)) = 0.00037

(b) Probability = 4*((1/4)*(1/32)3 ) = 0.000031

(c) Probability = 4! *((1/4)*(3/64)2 * (1/24)) = 0.00055

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