Considerate a game in which a coin will be flipped three. For each heads you will be paid $100. Assume that the coin comes up heads with probability 2/3.
A .Construct a table of the possibilities and probabilities in this game.
B. Compute the expected value of the game.
C. How much would you be willing to pay this game?
D. Consider the effect of a change in the game so that if tails comes up two times in row, you get nothing. How would our answer to the first three parts of this question change?
It is important to state that probability dictates there is no such thing as having a 2/3 chance of constantly tossing a coin and it landing on either heads or tails. You either have a dodgy coin or there is something else going wrong. Tossing a coin should always give the equal probability of getting either heads or tails.
Of course the tail is not a honest one as the probability of heads coming up is 2/3 and tails 1/3.
When you toss the coin thrice, the probability of you getting:
0 heads is (1/3)^3 = 1/27
1 head is 3*(2/3)*(1/3)^2 = 6/27
2 heads is 3*(2/3)^2*(1/3) = 12/27
and 3 heads is (2/3)^3 = 8/27
As you are paid $100 for a head, you are likely to be paid 0 + 600/27 + 1200/27 + 800/27 = 2600/27 = $96.29
The game can be played for any amount lower than this to make a profit.
For your last question just make the appropriate changes in probability in the above calculation to find the required values.
I was also wondering about the probability of the coin coming up heads. First, I would not be willing to risk more money than I could afford to lose (so today, I would not be playing). If I were to play though, I would begin with one dollar (assuming that the game costs a quarter a play). Hopefully, I would win one in four flips. After that win, I would be more willing to gamble more (with the winnings only).
First of all how can there be a 2/3 probability of the coin coming up heads? Is it a dishonest coin? If so, I could assume that I would win $200 in the average game. That means that I would pay less than $200 to play. I would probably pay a good deal less because I would A) know that I would not necessarily get the average game and B) because I am essentially investing what I pay for the hope of getting $200. I would need to leave a large margin to make the risk worth it.
To elaborate a bit on the first answer: if it cost me a quarter to play the game and I had a chance of winning $100 each time I bet a quarter, I would probably play this game until I couldn't play it any longer. I generally don't care to play games, but this is one I would really enjoy!
You have a pretty high chance of getting heads. You did not say how much it costs to play the game. In order to really answer the question, we need to know not only what you could win but what you are betting, or the cost to play the game.