Is this a fair game?

Question

This isn't a homework problem. I just feel like awarding ten points to some deserving person.

The game works as follows: After paying a $1 entry fee, the PLAYER tosses a coin. If it comes up heads, the HOUSE gives him $1 and the player gets to toss the coin again. The player continues to toss the coin, collecting $1 for each time it lands heads, until it comes up tails. When it lands tails, the player collects nothing and the game is over.

Clearly, the most the player can lose is $1, while the most he can win is unlimited. So, in this game, which is true: (a) the player has a statistical advantage, (b) the house has a statistical advantage, or (c) the game is fair - no statistical advantage to either the player or the house. Naturally, to get the points, you need to show the math to prove your answer.

Answer 1

The number of times you have to flip a coin before you get a tail (a random variable I'll call X) has an geometric distribution with probability of success p = 1/2. The expected value of the geometric is E(X) = 1/p, which in this case is 2, meaning that the number of coin flips it takes to get a tail is 2 on average. Let's have the random variable W be the winnings.

Now if X = 1, then your winnings W = -1. (You gave the house a dollar, but you got a tail right away, so you didn't win anything. Net is -1.)
If X = 2, then your winnings is W = 0. (You gave the house a dollar, and you got a head first and tail second. Net is 0.)
If X = 3, then your winnings is W = 1. (You gave the house a dollar, and you got head, then head, then tail. Net is 1.)

And so on. So it looks like W = X - 2.

So the expected winnings is E(W) = E(X-2) = E(X) - 2 = 2 - 2 = 0, meaning that you would expect to neither win nor lose money playing this game.

The game is fair. Option c.

Answer 2

My answer would be: it depends. If the player only has the original $1 to play with, he has a 50% chance of winning it back the first time he plays. From a purely economic standpoint, the HOUSE has an advantage on this first game, because even if the PLAYER wins, the HOUSE loses nothing -- the PLAYER just gets his $1 back -- but statistically it's a fair game. Each subsequent toss of the coin is also a fair game, but economically, after the second win, the advantage is to the PLAYER, because he not only has his original $1 back in his possession, but he's ahead of the HOUSE. Even if he loses, he still has more than he started out with.

Overall, however, the game leans in the HOUSE's advantage, because for the PLAYER to come out ahead, he has to get at least two heads in a row, which is 1/2 X 1/2 = 25% chance. Pretty decent odds, as far as gambling goes, but it still means the HOUSE has a 75% chance of at least keeping its money, and a 50% initial chance of taking the PLAYER's $1 away from him.

Answer 3

The game is obviously unfair in the players favor.

If the game were to end after one cycle, it would be fair and even. Since the player is not charged an addtional dollar for subsequent tosses after a winning toss, the player has an advantage during those turns.

Answer 4

Expected Value = P(t)*(-1) + p(h,t)*0 + p(h,h,t)*1 + p(h,h,h,t)*2 + ...

=.5*(-1) *.25*(0) + 1/8 + 2/16 + 3/32 + 4/64 + 5/128 + 6/256...

which equals -.5 + 0 + .5 (as the series goes to infinity) = 0

The game is precisely fair

Answer 5

Entry fee is $1.
Odds of breaking even.....50%
Odds of winning....50%
Odds of losing and ending game....50%

The answer is C, since there is no statistical advantage to either the player or the house. Although, since the player has to first pay $1 for entry, then you could say that the house has the advantage, because if the player wins on his first attempt, all he suceeds in doing his regaining his entry fee.

Answer 6

The game is in the players favor and the more he plays the better his earnings will be.

The player has a 50 percent chance of loosing the dollar
a 25 percent chance of winning one dollar,
12.5 percent chance of winning 2 dollars,
6.25 percent chance of winning 3 dollars,
3.125 present chance of winning 4 dollars, etc.

If you look at the odds of winning versus loosing its 50/50 but if you look at how much he will profit then it is in his advantage.

Answer 7

c- fair game because its a 50-50 chance that he will land on heads or tails