4. John plans to organize a gift exchange at a party. All the gifts will be put

Question

4. John plans to organize a gift exchange at a party. All the gifts will be put into identical color

boxes so no one can tell whose gift is in a box. The boxes will be randomly distributed to the

partygoers. It is not desirable for anyone to be assigned his/her own gift. John wants to find out

what the probability is that no one is assigned his/her own gift after the exchange.

(a) Suppose there are 4 people attending the party. In how many ways can the boxes be distributed

so that exactly one person is assigned his/her own gift?

(b) Suppose instead that there are 5 people attending the party. In how many ways can the

boxes be distributed so that no one is assigned his/her own gift? [Hint: You may want to

start with a party of only 4 people and consider the complementary event. Then extend your

reasoning to the case of 5 people.]

(c) If the boxes are randomly distributed in a party of 5, what is the probability that no one is

assigned his/her own gift?

(d) John knows that his best friend Galerius, one of the other 5 people attending the party, gives

exceptionally good gifts. However, John does not feel as confident in his own gift. What

is the probability that John ends up receiving Galerius’ gift, but Galerius does not receive

John’s gift?

Thanks.

Explanation / Answer

Solution :-

a) Suppose there are 4 people attending the party. In how many ways can the boxes be distributed so that exactly one person is assigned his/her own gift?

If there was an exact match, then the other 3 people can only pick 2 other gifts(that is the one which does not match them), so it would be 1*2*2*2 = 8

b) Suppose instead that there are 5 people attending the party. In how many ways can the boxes be distributed so that no one is assigned his/her own gift? [Hint: You may want to start with a party of only 4 people and consider the complementary event. Then extend your reasoning to the case of 5 people.]

Since it is no match with their own gifts, therefore the total number of ways minus the ways with an exact match with 5 people. Therefore, it shoud be 44

c) If the boxes are randomly distributed in a party of 5, what is the probability that no one is assigned his/her own gift?

P(no one is assigned his/her own gift) = 44 / 120 = 0.37

d) John knows that his best friend Galerius, one of the other 5 people attending the party, gives exceptionally good gifts. However, John does not feel as confident in his own gift. What is the probability that John ends up receiving Galerius’ gift, but Galerius does not receive John’s gift?

P (John ends up receiving Galerius’ gift, but Galerius does not receive John’s gift) = 24/720 = 0.033

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