For a bet on a coin toss with a fair coin where P(Heads) = P(Tails) = 0.5. In this example, I am the one accepting the bets. I.e. I"m the 'house' so to speak. I pay out even money on a winning bet. So if you bet \$10, you make \$10 profit.
If you place a \$10 bet and you lose, I keep the \$10. If you place a \$10 bet and you win, I give you \$20 - that is, $10 winnings plus your \$10 stake.
What is my expected payout on any given \$10 bet that I accept? Is it a) \$20 * 0.5 = \$10 or is it b) 10 + (10*5) = \$15?
The idea with b) being that no matter what the odds are, you will always get the full stake back on a winning bet so you don't apply the probability to the stake - it is a constant. As odds change, the only thing that changes is your winnings.
Prob winning = 0.5. Odds = 2. Bet amount, \$10. Actual winnings = \$10. Total payout = \$20. Expected winnings = 10*.5 = \$5. Expected Payout= \$15
Prob winning = 0.526315789. Odds = 1.9. Bet \$10. Actual winnings = \$9. Total payout = \$19. Expected winnings = 9 * 0.526315789 = 4.736842105. Expected Payout= \$14.736842105
Which approach is correct and why (or why is one wrong)? As the 'house' how much do I expect to pay out on a $10 bet with a 0.5 chance of winning?