The expected number of draws until four successive cards of the same suit appear

Question

This is question 3.24 from Stochastic Process, 2nd ed. by Sheldon Ross.

Draw cards one at a time, with replacement, from a standard deck of playing cards. Find the expected number of draws until four successive cards of the same suit appear.

I know if I want 4 successive cards of the same suit given the suit the expectation would just be $4^4+4^3+4^2+4$. But I don't know how to proceed if I am not given the suit.

Any help would be appreciated. Thanks!

Answer 1

We have the equalities:

• $\mu_{3}=1+\frac{3}{4}\mu_{1}$
• $\mu_{2}=1+\frac{3}{4}\mu_{1}+\frac{1}{4}\mu_{3}$
• $\mu_{1}=1+\frac{3}{4}\mu_{1}+\frac{1}{4}\mu_{2}$
• $\mu_{0}=1+\mu_{1}$

Here e.g. $\mu_2$ stands for the expectation under the extra condition that from the cards drawn already the last $2$ cards (and not more) are of the same suite.

Note that in that situation there is a probability of $\frac14$ that we arrive in the situation where the last $3$ cards drawn are of the same suite and a probability of $\frac14$ that a card of another suite will be drawn so that we must start over again.

This consideration makes clear that $$\mu_2=\frac34(1+\mu_1)+\frac14(1+\mu_3)=1+\frac34\mu_1+\frac14\mu_3$$

To be found is $\mu_0$ and the equalities enable you to find it.