Question

You now have $4000. You will toss a fair coinup tofour times. Before each toss you can bet any amount of your money (including none) on the outcome of the toss. If heads comes up, you win the amount you bet. If tails comes up, you lose the amount you bet. Your goal is to reach $6000. It turns out that you can maximize your chance ofreaching$6000 by betting either the money you have on hand or $6000minus the money that you have on hand, whichever is smaller. Use simulation to estimate the probability that you will reach your goal with this betting strategywith the initial budget $4000.

Answer 1

Since we want to bet the smaller amount, $6000 - budget, each time, we will be betting $2000 dollars each time. We know the odds of landing on heads with a coin is a %50 chance. Now if we get heads we can win. If we do not get heads we are left with $2000. Follow the betting equation again. $6000 - $2000 = $4000. We do not have $4000 so we will bet $2000. Once again we have a %50 chance of winning. If we win we will have $4000. If we loose, we will be left with nothing. If we were to loose, we would no longer be able to reach our goal because lack of money. Now lets say we won. We would have $4000 again so we would have to repeat like before. We would bet $2000 and if we win, we reach the goal. If we lost, we would also have to repeat steps before. This means the game could be over in 1 flip but could go on forever.