The students in a class decide to send a representative to the president of scho

Question

The students in a class decide to send a representative to the president of school to complain about their terrible classroom. But nobody in the class will agree to be the representative, so they decide on the following strategy: Each student will toss his/her coin simultaneously; if all the coins but one show the same face, the person who tossed the odd coin must be the representative. If any other combination of heads and tails occurs, the class will toss coins again.

(a) Assuming that there are n students in the class and that all of the coins are fair, what is the expected number of rounds (number of coin tosses each student has to make)? (Hint: Explain why the probability of an odd toss in any given round is p = n/2 n1 . Then explain why the probability of ending on the kth round is (1 p) k1p. Finally, using the method of Example 25 on page 167 of Rosen, get the answer 2n1/n.)

(b) One of the n students decides to cheat and uses a false coin that comes up heads much more often than tails. How does this change the answer in part (a)? Explain

Explanation / Answer

Solution:

Since a coin has only two outcomes,

so when a person tosses a coin the outcome of a particular face is 2 with a probability of 1/2

when two people are tossing the coin simultaneously then, 4 outcomes, HH, HT, TH, TT

H=> Head

T=> Tail

This means the number of possibilities is 2^(number of people tossing the coin)

Now if n people are tossing the coin then number of possibilities are 2^n

the probability of 1 getting the one person an odd face will be= 1/2^(n) - 1

so, the probability of n getting the one person an odd face will be= n/2^(n) - 1

b)

If one naughty has a faulty coin which scores head more often than tail then it is highly likely that the odd face will be a head one, in this case other (n-1) coins will be behaving normally

so the number of possibilities for them will be 2^(n-1)

then the probability will be= (n-1)/ 2^(n-1)

I hope this helps if you find any problem. Please comment below. Don't forget to give a thumbs up if you liked it. :)

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