New Year's Countdown Day 24: Naughty or Nice

Probability Level 3

Three children - Nate, Olivia, and Patrick - are waiting for their presents to arrive from Santa. Santa put two of these children on the naughty list, and they will only receive a lump of coal on Christmas. The remaining child was put on Santa's nice list, so he/she will get a large gift on Christmas. However, none of the children know who was put on the nice or the naughty list, so they are waiting anxiously for Christmas to arrive so that they can find out.

On Christmas Eve, Nate is visited by a holiday spirit, who says that it has information on who Santa put on his naughty and nice lists. When Nate asks the spirit who among the three children is the nice one, the spirit replies, "I cannot tell you whether you were put on the naughty or nice list, but I can say truthfully that Olivia is one of the naughty ones." Before Nate can ask the spirit another question, it vanishes into thin air, leaving Nate to wonder if he is still naughty or not.

Can you help Nate out? What is the probability that Nate is on the naughty list, given the information from the spirit?

Details and Assumptions

Assume that if Nate were the nice child, then the spirit will choose to say either Olivia or Patrick is naughty with equal probabilities.

This problem is part of the set New Year's Countdown 2017 .

\dfrac{2}{3}

\dfrac{1}{2}

\dfrac{1}{4}

1

None of these choices

\dfrac{1}{3}

0

\dfrac{3}{4}

1 solution

Steven Yuan
Dec 28, 2017

This problem is equivalent to Monty Hall's problem, except with the goats replaced by "naughty" and the car replaced by "nice." If you are familiar with Monty Hall, then the answer will make sense. If not, here's a simple explanation for why the answer is $\dfrac{2}{3}$ and not $\dfrac{1}{2},$ as many might believe:

At first, when Nate has no information about who is on the naughty or nice list, he has a $\dfrac{2}{3}$ chance of being placed on the naughty list. When the spirit says that Olivia is on the naughty list, this does not change the fact that originally, Nate had a $\dfrac{2}{3}$ chance of being naughty. This is the same as saying that, for the original Monty Hall problem, you'll get a goat with probability $\dfrac{2}{3}$ if you selected a goat for your first choice and decided to stay with your selection. The reveal of the other goat does not change this probability.

Therefore, Nate has a $\boxed{\dfrac{2}{3}}$ chance of being on the naughty list.

New Year's Countdown Day 24: Naughty or Nice

This problem is part of the set New Year's Countdown 2017 .

1 solution

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