Gambler is deciding whether or not to take a bet. she must pay $40 to take the bet, but if she wins, she willprofit$225.theprobability that she wins the bet is ¼. what is the player’s expected value in this situation

Expected value of the bet is
the sum of the products of value of outcome and its probability,
less the amount paid to place the bet.

Outcomes value probability
win            225    1/4
lose               0    3/4
cost of bet = 40

So expected value of bet
E[X]=225*(1/4)+0*(3/4)-40
=56.25-40
=16.25

This means that in the long run, gambler will win, since the expected value is positive.  (does NOT mean she will win in the next bet!)