what are the expected returns for these types over the long run if i win 50% of legs on each of these types of parlay bets
Type 1 - 8 legs, 80x for 8/8, 3x for 7/8, 1x for 6/8
Type 2 - 7 legs, 40x for 7/7, 2.5x for 6/7, .5x for 5/7
Type 3 - 6 leg, 25x for 6/6x, 2.6x for 5/6, .25x for 4/6
Type 4 - 5 leg, 10x for 5/5, 2.5x for 4/5
Type 5 - 5 leg, 20x for 5/5
Type 6 - 4 leg, 6x for 4/4, 1.5x for 3/4
Type 7 - 4 leg, 10x for 4/4
Type 8 - 3 leg, 6x for 3/3
Type 9 - 3 leg, 3x for 3/3, 1x for 2/3

To calculate the **expected returns** for each type of parlay bet, we need to compute the expected value (EV) using probabilities. Since you win **50% of legs** on average, the probability of correctly guessing a certain number of legs follows a **binomial distribution**.

Let’s denote the following terms:  
- \( N \): Number of legs in the parlay.
- \( k \): Number of correct legs.
- \( P \): Probability of correctly predicting a leg = 0.5.
- \( C(N, k) \): Number of ways to choose \( k \) correct legs from \( N \) = \( \binom{N}{k} \).
  
The **general formula for expected value (EV)** is:

\[
\text{EV} = \sum_{k = X}^{N} \left( \text{payout for } k \text{ correct legs} \times \Pr(k \text{ correct legs}) \right)
\]

where \( \Pr(k \text{ correct legs}) = C(N, k) \cdot P^k \cdot (1 - P)^{N - k} \).  
We’ll now calculate the EVs

what are the expected returns for these types over the long run if i win 50% of legs on each of these types of parlay bets Type 1 - 8 legs, 80x for 8/8, 3x for 7/8, 1x for 6/8 Type 2 - 7 legs, 40x for 7/7, 2.5x for 6/7, .5x for 5/7 Type 3 - 6 leg, 25x for 6/6x, 2.6x for 5/6, .25x for 4/6 Type 4 - 5 leg, 10x for 5/5, 2.5x for 4/5 Type 5 - 5 leg, 20x for 5/5 Type 6 - 4 leg, 6x for 4/4, 1.5x for 3/4 Type 7 - 4 leg, 10x for 4/4 Type 8 - 3 leg, 6x for 3/3 Type 9 - 3 leg, 3x for 3/3, 1x for 2/3

Learn how to calculate expected returns for parlay bets with different legs, payout structures, and probabilities. This example uses binomial distribution and expected value calculations to assess long-term returns when winning 50% of legs. Suitable for advanced high school students.

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