The house edge for parlays:
In general, each wager has a 50% or ½ probability of winning. If we are trying to determine the probability of hitting multiple bets, we just multiply ½ by the number of bets we are making. The probability of hitting a two team parlay is ½ by ½ = ¼. Another way to express this is to way that we will lose 3 times for each time that we win. Still another way to express this is to say that the fair odds are 3-1. A normal payout for a two team parlay is 13-5. Assuming a $5 wager, we can calculate the house edge as follows.
Expectation = [1/4 x (+13)] + [3/4 x (-5)]= -2/4= -0.5
E per $1 bet = -0.5/5 = -0.10, or 10% house advantage
For a six team parlay, the true probability of winning is ½ x ½ x ½ x ½ x ½ x ½ = 1/64. This means that we can expect to lose 63 times for each time that we win (fair odds of 63-1). If the casino pays out at 44-1 odds (typical payout), the house edge for the bet is:
E=[1/64 x (+44)] + [63/64 x (-1)] = -19/64 = -0.297, or 29.7% house advantage
The house edge for straight bets:
Assuming the same 50% win probability as for parlays and using an $11 bet (the house pays 10-11 on straight wagers), we can confirm the house edge.
Expectation=[0.5 x (+10)] + [0.5 x (-11)]= -0.5
E per $1 bet = -0.5/11 = -0.045, or 4.5% house advantage
Overcoming the house advantage for straight bets:
Straight bets are made risking $11 to win $10. To overcome the house advantage you will have to win 11 times for every 10 times you lose.
[10 x (-11)] + [11 x (+10)] = 0
An easier way to express this is to say that you have to win 11 out of 21 bets, 11/21=0.5238, or 6=52.4% to break even.