# Parlez Vous Parlay?

This past week, I received a most intriguing request from a reader.  He asked me to review 'again' the true odds associated with parlay cards.  I found the 'again' quite amusing as to the best of my knowledge this is not a topic I have ever evenly remotely covered.  At first, I couldn't even figure what I would write about.  Then, after giving it more thought, I realized that there might be some math information that parlay players would like to think about.  I apologize that I'm waiting until after the football season to present this information.

The first thing that you need to understand about any sports wager is that the goal of the sportsbook is to get half of the money on each side of the wager.  Since a straight wager generally pays 10 to 11 (meaning you wager \$11 to win \$10), if the money is evenly split between the two sides of a wager, they're going to make money.  They way they control this is by adjusting the line of the wager - in a sport like football this is the point spread or the over/under line.  If a sportsbook moves the point spread from +5 to +6, it doesn't really mean that they think one team is another point better than the other.  It means that they need to more money on the underdog.  They still have some risk that the final score could be very close to that 5 or 6 point spread and things can get ugly in that case.  But, they use very sophisticated computer programs to help them manage the risks, so don't cry for the sportsbooks.

In the end, we pretty much have to assume that the probability of picking the winner of a game  including the point spread or picking the over/under is essentially 50%.  If you have some inside knowledge of injuries or the like, obviously, you can affect that.  But, for most of us mortals, it's going to be a 50/50 shot.  Given this, calculating the 'true odds' of a parlay is a simple as multiplying 0.50 by itself for each additional line on your parlay.  So, the odds of picking 1 out of 1 is 50%, 2 out of 2 is 25%, 3 out of 3 12.5%, etc...  Flip these around to get the payout you'd like to see for a 100% payout.  The table below shows, these numbers:

Parlay Lines           Probability     True Odds Line

2 out of 2                 25.00%                4 for 1

3 out of 3                 12.50%                 8 for 1

4 out of 4                  6.25%                16 for 1

5 out of 5                   3.13%                32 for 1

6 out of 6                  1.56%                64 for 1

7 out of 7                  0.78%              128 for 1

8 out of 8                  0.39%             256 for 1

So, all things being equal, if you go for a 4-line parlay, you'd love to get paid 16 for 1 (or 15 to 1, which is the same).  You're probably not going to get anywhere near that.  If you're lucky, probably 11 or 12 to 1.

As I said earlier, I wasn't sure how much there was to write about this topic.  But, then I remember those 'Teaser' parlays and the 'Ties Win' parlays.  Most parlays use 1/2 points to avoid any ties.  Some will simply remove a tie from the equation.  So if you pick 4 lines and one ends a tie, it is as if you picked a 3 line parlay with the other lines.   With a Ties win parlay, your odds of winning any line (and thus the parlay) goes up.  There is no way to determine by how much.  If I had a database of all games available to me, I might be able to figure out how often a tie (with the point spread) actually occurs.  Since I don't have this, I'll have to give you some estimates.  So, I'll assume two possibilities for the Ties win parlay - that your probability to win is 52% or 55%.

With the Teaser parlay, the point spread is increased in both directions so that the probability that you can pick any particular line increase significantly.  Again, I don't have hard numbers to know by how much, so I'll use 60 and 65%.  The tables below show the probability and true payout for a 100% payback for each of these probabilities:

Ties Win

52%                                              55%

Parlay Lines    Probability   True Odds Line     Probability    True Odds Line

2 out of 2           27.04%           3.70 for 1               30.25%          3.31 for 1

3 out of 3            14.06%             7.11 for 1               16.64%          6.01 for 1

4 out of 4               7.31%         13.68 for 1                  9.15%       10.93 for 1

5 out of 5             3.80%         26.30 for 1                  5.03%       19.87 for 1

6 out of 6             1.98%          50.58 for 1                  2.77%       36.13 for 1

7 out of 7             1.03%          97.27 for 1                   1.52%       65.68 for 1

8 out of 8            0.53%        187.06 for 1                  0.83%      119.43 for 1

Teaser

60%                                            65%

Parlay Lines     Probability    True Odds Line     Probability   True Odds Line

2 out of 2            36.00%            2.78 for 1              42.25%        2.37 for 1

3 out of 3             21.60%            4.63 for 1              27.46%        3.64 for 1

4 out of 4             12.96%            7.72 for 1               17.85%        5.60 for 1

5 out of 5               7.78%          12.86 for 1                11.60%        8.62 for 1

6 out of 6              4.67%          21.43 for 1                  7.54%      13.26 for 1

7 out of 7              2.80%          35.72 for 1                 4.90%     20.40 for 1

8 out of 8               1.68%          59.54 for 1                  3.19%      31.38 for 1

You'll note that as the probability of picking any one game correctly goes up, the probability of picking multiple games correctly goes up even more.  So, as the probability goes from 55% to 65%, the probability of 8 out of 8 increase nearly four fold.  Accordingly, the expected payout gets cut by a factor of 4.

If you want to know the real payout you are getting, multiply the probability by the actual payout.  Of course, you have to decide what the probability of you picking a game correctly is.  There's a good reason why my father used to always joke to not bet on any event where the participants have two legs!  Good luck!