Siuan's grandmother recently signed up for a twitter account. She only follows five people - her five grandchildren. Each day, each of her grandchildren make two tweets and at the end of the day Siuan's grandmother gets an e-mail that lists all ten tweets in chronological order. If the times of each grandchild's tweets are random, the probability that no consecutive pair of tweets in the e-mail are by the same person can be expressed as a/b where a and b are coprime positive integers. What is the value of a + b?
2 answers
fail.
First, we will find the probability of at least one grandchild making consecutive tweets by using the inclusion-exclusion principle, and then subtract the result from 1.
From the inclusion-exclusion principle,
P(at least one grandchild makes consecutive tweets)
= (5 choose 1)P(a specified grandchild makes consecutive tweets) - (5 choose 2)P(two specified grandchildren makes consecutive tweets) + (5 choose 3)P(three specified grandchildren makes consecutive tweets) - (5 choose 4)P(four specified grandchildren makes consecutive tweets) + (5 choose 5)P(all five grandchildren makes consecutive tweets).
For 1 <= k <= 5, we need to find the probability that k specified grandchildren (and possibly others) make consecutive tweets.
There are 10! permutations of all 10 tweets.
Think of arranging k "blocks" of two tweets each, and (10 - 2k) single tweets. There are (10 - k)! ways of arranging these (10 - k) items, and 2 ways of arranging the two tweets within each of the k blocks. So (10 - k)!(2^k) of the 10! permutations result in k specified grandchildren (and possibly others) making consecutive tweets.
So the probability that k specified grandchildren (and possibly others) make consecutive tweets is (2^k)/[10*9*...*(10 - k + 1)].
Therefore, we have
P(at least one grandchild makes consecutive tweets)
= 5(2/10) - 10(4/(10*9)) + 10(8/(10*9*8)) - 5(16/(10*9*8*7)) + 1(32/(10*9*8*7*6))
= 1 - 4/9 + 1/9 - 1/63 + 1/945.
So the probability of no grandchild making consecutive tweets is
1 - (1 - 4/9 + 1/9 - 1/63 + 1/945) = 4/9 - 1/9 + 1/63 - 1/945 = 47/135.
From the inclusion-exclusion principle,
P(at least one grandchild makes consecutive tweets)
= (5 choose 1)P(a specified grandchild makes consecutive tweets) - (5 choose 2)P(two specified grandchildren makes consecutive tweets) + (5 choose 3)P(three specified grandchildren makes consecutive tweets) - (5 choose 4)P(four specified grandchildren makes consecutive tweets) + (5 choose 5)P(all five grandchildren makes consecutive tweets).
For 1 <= k <= 5, we need to find the probability that k specified grandchildren (and possibly others) make consecutive tweets.
There are 10! permutations of all 10 tweets.
Think of arranging k "blocks" of two tweets each, and (10 - 2k) single tweets. There are (10 - k)! ways of arranging these (10 - k) items, and 2 ways of arranging the two tweets within each of the k blocks. So (10 - k)!(2^k) of the 10! permutations result in k specified grandchildren (and possibly others) making consecutive tweets.
So the probability that k specified grandchildren (and possibly others) make consecutive tweets is (2^k)/[10*9*...*(10 - k + 1)].
Therefore, we have
P(at least one grandchild makes consecutive tweets)
= 5(2/10) - 10(4/(10*9)) + 10(8/(10*9*8)) - 5(16/(10*9*8*7)) + 1(32/(10*9*8*7*6))
= 1 - 4/9 + 1/9 - 1/63 + 1/945.
So the probability of no grandchild making consecutive tweets is
1 - (1 - 4/9 + 1/9 - 1/63 + 1/945) = 4/9 - 1/9 + 1/63 - 1/945 = 47/135.
1 answer