Three children, John, Paul, and Ringo, are playing a game.

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Three children, John, Paul, and Ringo, are playing a game.

by BTGModeratorVI » Sat Jun 27, 2020 6:37 am

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Three children, John, Paul, and Ringo, are playing a game. Each child will choose either the number 1 or the number 2. When one child chooses a number different from those of the other two children, he is declared the winner. If all of the children choose the same number, the process repeats until one child is declared the winner. If Ringo always chooses 2 and the other children select numbers randomly, what is the probability that Ringo is declared the winner?

A. 1/6
B. 1/4
C. 1/3
D. 1/2
E. 2/3

Source: Princeton Review

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Re: Three children, John, Paul, and Ringo, are playing a game.

by [email protected] » Tue Jun 30, 2020 5:58 am
BTGModeratorVI wrote:
Sat Jun 27, 2020 6:37 am
Three children, John, Paul, and Ringo, are playing a game. Each child will choose either the number 1 or the number 2. When one child chooses a number different from those of the other two children, he is declared the winner. If all of the children choose the same number, the process repeats until one child is declared the winner. If Ringo always chooses 2 and the other children select numbers randomly, what is the probability that Ringo is declared the winner?

A. 1/6
B. 1/4
C. 1/3
D. 1/2
E. 2/3

Source: Princeton Review
This is one of my all-time favorite questions!!

The main concept here is that all 3 children are equally likely to win this game (unless one of them possesses supernatural powers that allow him to know what numbers the other two boys will choose )

Also note that, if everything is random, the probability of winning by choosing the number 2 is the same as the probability of winning by choosing the number 1.

So, regardless of what number Ringo chooses, his probability of winning is exactly the same as each of the other boys winning.

Since all 3 boys have the same probability of winning, P(Ringo wins) = 1/3

Likewise, P(John wins) = 1/3 and P(Paul wins) = 1/3

Cheers,
Brent

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Re: Three children, John, Paul, and Ringo, are playing a game.

by [email protected] » Mon Apr 19, 2021 6:11 am
BTGModeratorVI wrote:
Sat Jun 27, 2020 6:37 am
Three children, John, Paul, and Ringo, are playing a game. Each child will choose either the number 1 or the number 2. When one child chooses a number different from those of the other two children, he is declared the winner. If all of the children choose the same number, the process repeats until one child is declared the winner. If Ringo always chooses 2 and the other children select numbers randomly, what is the probability that Ringo is declared the winner?

A. 1/6
B. 1/4
C. 1/3
D. 1/2
E. 2/3

Solution:

Since Ringo always chooses the number 2, there are only 4 possible outcomes for any round of the game. We list Ringo’s choice first, followed by John’s and then Paul’s:

(2, 1, 1) Ringo wins

(2, 1, 2) Ringo loses (John wins)

(2, 2, 1) Ringo loses (Paul wins)

(2, 2, 2) No one wins, so another round is played. When this happens, the same 4 outcomes occur for the next round and for subsequent rounds.

We see that in any given round, each boy has a 1/3 chance of winning. It doesn’t matter if Ringo always chooses 2 or just randomly chooses a number, his probability of winning is 1/3.