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