Four to a Royal Flush

EXAMPLE 1 – Keep a pair or chase a straight and Royal Flush?

Suppose we are dealt Kc Qc Jc Tc Kd. Do we hold a pair of kings or do we discard the Kd? After analyzing the two alternatives, our answer is easy. The schedule below shows the EV of our two alternatives.

The EV was calculated by dividing the sum of the expected payouts by the total number of combinations. In the long run, the player will make ~98 coins per hand when he or she holds the Kc, Qc, Jc, and Tc. On the other hand, holding the pair of kings nets the player ~8 coins per hand. Our math reveals that holding on to the Kc, Qc, Jc, and Tc is the correct play.

We’ll provide an alternate EV breakdown of throwing away the Kd and holding the Kc, Qc, Jc, and Tc. It’s very easy to calculate since we are only drawing one card.

First, there is one way to get a royal flush – the Ac. Here is the EV calculation:

(1 / 47) * 4,000 = ~85.11 coins

We can also get a straight flush by drawing the 9c:

(1 / 47) * 250 = ~5.32 coins

If we keep the Kc, Qc, Jc, and Tc, it’s impossible for us to make four of a kind or a full house.

However, there are seven cards (2c, 3c, 4c, 5c, 6c, 7c, 8c) that complete a flush.

(7 / 47) * 25 = ~3.72 coins

Six cards (Ad, As, Ah, 9d, 9h, 9s) give us a straight:

(6 / 47) * 20 = ~2.55 coins

Finally, eight cards (Kh, Ks, Qh, Qs, Qd, Jh, Js, Jd) will gives us a qualifying hand of Jacks or Better.

(8 / 47) * 5 = ~0.85

Sum up all the EV calculations and you will get ~97.55 coins – the expected value of discarding the Kd and holding the Kc, Qc, Jc, and Tc.

Here is the EV of holding the pair of kings and drawing three cards: