Solution to Probability to Win the $5000

For each faculty member by seniority

You and four colleagues find a $5000 bill on the floor of your department’s faculty lounge. You ask around but nobody seems to have any idea where it came from. None of you have change, so you agree to play a game of chance to decide who keeps it. The five of you sit around a table and the bill is placed in front of you as you are the most senior faculty member. The other faculty sit with the next senior person on your right then the next and so on. The game is played in turns. Each turn, the person in front of whom the bill is will throw a six-sided die. If any number from 1-4 shows up, (probability 2/3) the bill moves to the right one position (i.e. towards the next junior person). If 5 or 6 show up (probability 1/3) the bill stays put and the person who is sitting in front of the bill gets to keep it. The bill starts in front of you, and you get to throw first. What are the chances you win the money? And what are your colleagues’ chances in order of seniority to win the coin?

Solution:

We will need the formula for the Geometric series:

S = a + ar + ar² + ar³ + ar⁴ +…

rS = ar + ar² + ar³ + ar⁴ +ar⁵ + …

S(1-r) = a

Hence, S = a/(1-r)

Let us label the faculty by seniority S1,S2, S3, S4, S5 with S1 (you) being the senior most. On each throw, p =1/3 is the probability that the bill stays where it is and q = 2/3 that it moves to the right (to the next junior person). You will win at the first try with probability p. If you do not win on the first try (with probability q), then you will win if the bill keeps moving on the next four tries (probability q­­⁴) and you win on the sixth try. This sequence happens with probability q⁵­p Then you win again on the 11 th try with probability q¹⁰p and so on. So, the probabilities are:

S1 (you) wins = p + q⁵p + pq¹⁰+ … = p (1+ q⁵ + q¹⁰ + …) = p /(1-q⁵) ~ 0.384.

S2 wins = qp + qpq⁵ + qpq¹⁰+ … = qp /(1-q⁵) ~ 0.256

S3 wins = q²p + q²p q⁴ + q²p q⁸+ … = q²p/(1-q⁵) = 0.171

S4 wins = q³p/(1-q⁴) ~ 0.113

S5 wins = q⁴p/(1-q⁴) = 0.076

These probabilities must add up to 1.

Sum = (p + qp + q²p + q³p + q⁴p) /(1-q⁵) = p/(1-q) = 1.