What is the expected gain from taking a winner's bet? Suppose a gambler makes a series of independent bets. In each bet, there's a 50% chance that he loses $\$100$.
If he wins, he has an option to take $\$100$ or run a "winner's bet", instead of taking that reward. In the winner's bet, there is a 60% chance to win 50 dollars, a 30% chance to win 150 dollars, and a 10% to win 200 dollars.
What should he do in order to maximize the expected reward?
Note: This is not homework. I've been out of school for a decade. :)
 A: @Cardinal hasn't elaborated on his/her correct answer so I'll do so...
To learn the expected value of the whole we calculate the expected value of the parts.  Let's start with the optional winner's bet. 
Expected winnings = Value of each prize * the probability of each prize.
                    $\$$50 * 0.6 = $\$$30
                    $\$$150 * 0.3 = $\$$45
                    $\$$200 * 0.1 = $\$$20  
30+45+20 = $\$$95  
The winner's bet has expected winnings of $\$$95 but the gambler always has a choice between the winner's bet and a guaranteed $\$$100.  $\$$100 > $\$$95 so a gambler maximizing their expected winnings will never take the winner's bet.  Knowing that the winner's bet is a bad idea, we can ignore it when calculating the expected value of the overall game.  The game reduces to a 50% chance of winning $\$$100 and a 50% chance of losing $\$$100, resulting in expected winnings of $\$$0.
Therefore, the gambler can either play the game (and always decline the winner's bet) or not play the game at all, and have the same amount of money in expectation.
A: Agree with @Michael Bishop. Given the current probabilities on the winner's bet there is no incentive to opt for it. Given the your data, if the gambler opts for the winners bet his payoff on an average is 95, while his payoff is deterministically 100 if he does not opt for the winners bet.
These problems are best solved using a decision tree. Unfortunately I am not being allowed to post a picture showing the payoff choices for this problem. 
However, even a minor change in probabilities or the payoffs could change the choice the gambler needs to make to maximize his payoffs. You may play around with the numbers (probabilities / payoffs) and do some sensitivity analysis. For instance with either of the two changes mentioned below it would make more sense for the gambler to opt for the Winners bet.


*

*If the probability of wining 50, 150 and 200 were changed to 0.5, 0.25, 0.25 (keeping the payoffs same)

*If the maximum payoff is increased from 200 to 255, while the probabilities remained the same.

