Your coworkers are competitive people who like to gamble. They have set up a bet pool on the longest run of heads (call this R) that occurs in a sequence of 100 fair coin flips. In this pool you are allowed to bet on any combination of three mutually exclusive outcomes: 1) R < 6 2) 6 <= R < 10 3) R >= 10 Everyone has placed their bets and they are waiting for you to do the same. So far the money in the pool breaks down like this: $666 on outcome 1 $333 on outcome 2 $1 on outcome 3 for a total of $1000 in the bet pool. Assume your goal is to maximize your total expected value (as opposed to rate of return, or longterm growth rate). You have $200 in cash on you. Do you bet it all? Do you need to run to the ATM to get more? How much do you bet on each outcome? There is no "take" like at the race track. The whole $(1000 + your wager) is paid back to the winning bettors. Also, it doesn't matter how you arrive at the probabilities for the three outcomes, as long as they are close. It is only important that it is possible to calculate them as accurately as you like. This is not the case with a horse race. You can use .472, .484, and .044 as approximate values for outcomes 1, 2, and 3.