Suppose you and your friends are playing a game with a coin a. Suppose you bet $
ID: 3202977 • Letter: S
Question
Suppose you and your friends are playing a game with a coin
a. Suppose you bet $1 that the coin will land on heads, if you win you get a dollar (so you now have $) if you win again you get 2 dollars (so you now have a total of $4), if you win again you get 4 dollars (for a total of $8) and so on and so forth. If at any point you guess wrong you lose all your money. Assume the coin is a fair coin (P(heads)=P(tails)=.5) and you consistently bet on heads (so success is heads). Suppose your goal is to win more than a million dollars off of your friend. This requires you to win 20 times in a row. What is the probability that you will flip heads 20 times in a row?
b. Now suppose your friend is a little cheat, and he is using a (somehow) unfair coin. He knows you will always bet on heads. Suppose now the probability of getting heads is only .4. What is the probability of winning 20 times in a row now?
c. Instead suppose now that your friend wants to make a different bet. If he wins, you give him a dollar, and if you win he gives you $100. You win if, out of 12 tosses, exactly 10 are heads. What is the probability that you will win? Given these odds, is making this bet in your best interest (assume all you want to do is make money)?
d. Suppose the same scenario as before, but now assume you win if less than 10 of the 12 tosses are heads. Now is it in your best interest? (note, this is the same as saying you win so long as it isn’t 10, 11, or 12 heads)
Explanation / Answer
a)P(Flip head 20 times in a row)=0.5^20=9.54x10^-7
b)P(Win 20 times)=0.4^20=1.1x10^-8
c)P(I win)=12C10^0.5^12=0.0161
E(Profit)=100*0.0161-1*(1-0.0161)=0.6274
Hence I would be at an expected profit. Hence it would be a good attempt.
d)P(I win)=1-P(I lose)=1-P(10,11,12 heads)=1-(12C10+12C11+12C12)*0.5^12=0.9807
Hence E[Profit]=0.9807*100-1*0.0193=98.05
Hence it would be in my best interest