Question

2n random numbers are laid in a row. we take turns to take a number. each time, one can choose from the number on the tail or the head. the one with the bigger summation wins the game. if i go first, can i always win (or tie)?

Answer 1

The probability of having someone be picked four times in a row is 61/ 64 .

There are three possible outcomes on each draw: both of the same people from the previous round could be chosen, one of the same people could be chosen, or none of the same people could be chosen.

These have respective probabilities of

`[(1)/(8^2), (7)/(8) \cdot (1)/(8), (7)/(8) \cdot (6)/(8).]`

We only care about the first result, so we multiply and sum to find the expected outcome:

`[(1)/(8^2) + (7)/(8) \cdot (1)/(8) + (7)/(8) \cdot (6)/(8) = (61)/(256).]`

Then we multiply by 4 to account for having to do this 4 times and we conclude that the probability of having someone be picked four times in a row is 61/ 64 .

Question

If there were 8 of us playing a game and two people were picked at random every round, what are the odds one of those will be picked 4 rounds in a row?