Parlay Bets - Your Expected Winnings

Equation:
You calculate your expected winnings ("E") always like this:

E = Expected Winnings.
HE = House Edge, which is E x (-1). 
P = Probability to win from 0 to 1, where 0 is 'no chance' and 1 is 'absolutely certain'.
Pay = Payoffs, meaning money you get if you win. 

E = [P(win) x Pay(win)] + [P(loose) x Pay(loose)]

Parlay Payoffs & Probabilities:

The probability to win a parlay bet and the amount of money you'll get if you win is summarized in the following table:

No of teams picked	Probability to win	Payoffs
2	1/4	13:5
3	1/8	6:1
4	1/16	10:1
5	1/32	20:1

• Expected winnings for a Two Team Parlay Bet:
  The probability to win ("P(win)") is 1/4. 
  The probability to loose ("P(loose)") then is 1-P(win) = 1-1/4 = 3/4.
  The Payoff for winning the bet ("Pay(win)") is $13 for every $5 wagered.
  The Payoff for loosing the bet ("Pay(loose)") is -$5 then.

Hence,
=> E = [1/4 x $13] + [3/4 x -$5]
=> E = 13/4 - 15/4
=> E = -2/4 = -0.5

Your expected winnings for a $1 bet would be -0.5 / $5 = -0.1 or -10%.
Since your loosing is the house's winning to get the house edge you multiply your expected winnings with -1.
That's -10% x -1 = 10%.

• Expected winnings for a Three Team Parlay Bet:

=> E = [1/8 x $6] + [7/8 x -$1]
=> E = 6/8 - 7/8
=> E = -1/8 = -0.125
=> HE = -0.125 x -1 = 0.125 or 12.5%

• Expected winnings for a Four Team Parlay Bet:

  => E = [1/16 x $10] + [15/16 x -$1]
  => E = 10/16 - 15/8
  => E = -5/16 = -0.313
  => HE = -0.313 x -1 = 0.313 or 31.3%

   

• Expected winnings for a Five Team Parlay Bet:

  => E = [1/32 x $20] + [31/32 x -$1]
  => E = 20/32 - 31/32
  => E = -11/32 = -0.344
  => HE = -0.344 x -1 = 0.344 or 34.4%