Question

Question: A group of six people play the game of “odd person out” to determine who will buy refreshments. Each person flips a fair coin. If there is a person whose outcome is not the same as that of any other member of the group, this person has to buy the refreshments. What is the Probability that there is an odd person out after the coins are flipped once?

Short answer

Answer

The Probability that there is a one person out after the coins are flipped once is,$\frac{3}{16}$

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Step-by-step solution

Given Information

A group of six people play the game of “odd person out” to determine who will buy refreshments.

Each person flips a fair coin. If there is a person whose outcome is not the same as that of any other member of the group, this person has to buy the refreshments.

Definition of Probability of chance

A probability of chance is a number between 0 and 1 that shows how likely you are to get a certain result in a chance event.

Calculating the Probability

Given that six people play the game of “odd person out” for which each person flips a fair coin.

Hence, the Total number of outcome is as every coin has two possible outcomes i.e. Heads or Tails.

Given that the person whose outcome is not the same as that of any other member will be the odd man out, he is the only one who has to buy the Refreshments.

The number of Possibilities that there is an odd person is $C\left(6,1\right)$.

The possibility that each of them is different from all the others is when he gets a Head and all the others must get Tails and if he gets a Tails and all the others must get Heads.

That is only two possibilities out of the total $2^6$.

Now, the probability that there is an odd person out after the coins are flipped once is

$C\left(6,1\right)\times \frac{2}{2^6}=\frac{6}{32}=\frac{3}{16}$

Hence, the probability that there is an odd person out after the coins is flipped once is $\frac{3}{16}$.