Question

Yunsoo has ten cups face down and one cup has a $100 bill under it. You select cup 1. Yunsoo then groups cups 2-4 together into group A and cups 5-10 together into group B. He then flips all but one cup over in both groups to reveal that they do not have the $100 bill. What are your probabilities of winning for each of the remaining cups? Which cup should you . choose to have the highest chance of winning?

Answer 1

At first the probability of the $100 bill being under i-th cup was (1/10)=0.1 for all i=1,2,3,...10.

I select cup1 but it's not flipped and the probability of the $100 bill being under cup 1 is 0.1.

Now the probability of the $100 bill being in group A is (0.1+0.1+0.1)=0.3

and the probability of the $100 bill being in group B is (0.1+0.1+0.1+0.1+0.1+0.1)=0.6

Now Yunsooflip all the cups but one in each group and shows that those cups do not have the $100 bill.

Now only 3 cups are remaining the cup 1, the last cup of group A and the last cup of group B.

Anyone of the 3 has that $100 bill.

So the probability of wining for cup 1 is 0.1.

The probability of wining for the last cup of group A is

=P(the last cup of group A has the bill | the bill is in group A) P(the bill is in group A)

= 1 0.3 = 0.3

Similarly the probability of wining for the last cup of group A is

=P(the last cup of group B has the bill | the bill is in group B) P(the bill is in group B)

= 1 0.6 = 0.6

Hence I should choose the last remaining cup of the group B to have the highest chance of wining.