Joe Goes to the Casino!
Joe goes to a casino and encounters a new betting game. The dealer rolls a six-sided die. If Joe guesses which side will show, he wins $1000. But if he guesses wrong, he loses $200.
However, the die is not fair. Joe is told that the die is biased toward one side. More specifically, the probability that the die shows its biased side is 1/3, and the probability that it shows each of the other sides is 2/15. Joe assumes any of the sides is equally likely to be the biased one.
There is another catch. The dealer tells Joe that he can observe the die for a certain number of rolls. However, each observed roll costs $7. Joe must decide how many rolls to buy before playing (and he pays before playing). After observing the rolls he bought, Joe guesses which side will come up. Then the dealer rolls the die again, and Joe wins $1000 if he is right, and loses $200 if he is wrong.
Since Joe is rational, he decides to buy the number of rolls that maximizes the expected value of his earnings (which is what he wins minus the cost of buying the observed rolls). Call the number of rolls he buys N. Call his expected earnings E. N+E equals A/B where A and B are coprime positive integers. Find A+B.
Bonus 1 : Suppose Joe buys and observes 40 rolls. Does he expect to make any money?
Bonus 2 : What is the absolute maximum number of rolls Joe will be willing to observe (you can assume Joe is a risk-taker)?
Bonus 3 : What if Joe was risk averse and was only willing to place a bet if he knew he had at least a 40% chance of winning? What would Joe do then?
Bonus 4 : Given that Joe decides to buy and observe n rolls (n>=0), what is the probability that Joe is going to win? If Joe decides that he will only bet if he knows that he has at least a 30% chance of winning, what will Joe do then? (Note that even if he has at least a 30% chance of winning a bet, Joe will not place the bet if he expects to lose money.)
Bonus 5 : Assume Joe is allowed to play this game as many times as he wants. Can you come up with strategies that: maximize his long-term profit, or reduce his overall risk?
( Note : I am not 100% sure my answer is correct, so if you disagree with my answer, feel free to let me know!!)
The answer is 3097.
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1 solution
If I remember correctly, I got N = 5 and E = 2647/75 dollars.
Here's the Mathematica code I used:
Here's a plot of Joe's expected earnings E (x-axis is N):
NNmax = 7;
For[NN = NNmax, NN <= NNmax, NN++,
sumEV = 0;
listTuple = Tuples[{1, 2, 3, 4, 5, 6}, NN];
For[t = 1, t <= Length[listTuple], t++,
j = listTuple[[t]];
For[i = 1, i <= 6, i++, P[i, 0] = 1/6;];
P[i , m ] := (P[i,
For[i = 1, i <= 6, i++, n[i] = 0;];
For[m = 1, m <= NN, m++, n[j[[m]]]++;];
k = Ordering[{n[1], n[2], n[3], n[4], n[5], n[6]}, -1][[1]];
Pkbiased = P[k, NN];
Pknext = 1/3 Pkbiased + 2/15 (1 - Pkbiased);
sumEV = sumEV + (1000 Pknext - 200 (1 - Pknext));
]
f[NN] = sumEV/6^NN;
]
DiscretePlot[f[nn], {nn, 1, NNmax}]