Before we get started, congrats to Joseph Youngblood and Tripta, whose correct solutions to last month’s puzzles were no joke.

Main puzzle

To ensure that at least one of Al, Jan, Woo, or Sam hear the joke, we need 1 or more of the people in the middle layer to tell it. If neither of the people in the middle layer tell it, then Al, Jan, Woo, and Sam can’t hear it.

This can happen either because

• Cody doesn’t tell the joke to either of them,
• Cody tells the joke to one of them, but that person don’t tell it, or
• Cody tells the joke to both of them, and they both don’t tell it.

The total probability that no one among Al, Jan, Woo, or Sam hears it is the sum of these probabilities, which we can calculate in turn.

• The probability of the first scenario is $(1-p)^2$.
• The probability of the second scenario is $2p(1-p)^3$ since Cody has to not tell the joke to one person (probability $1-p$), tell the joke to the other person (probability $p$), and that person has to not tell it to either of the people they see (probability $(1-p)^2$). Finally there are two ways for Cody to pick who to tell.
• The probability of the third scenario is $p^2(1-p)^4$ since Cody has to tell the joke to both people he sees, who in turn have to both not tell the joke to either of the people they see.

Adding these up, and plugging in the quality of the joke $p = 0.6$, we get $24.6\%$. The probability that at least one of them hears it is $1$ minus this probability, or $75.4\%$.

Bonus question

In the infinite jokers situation, there needs to be at least one chain of joke tellers that lasts unbroken from Cody at the top all the way to the bottom of the social network.

Putting aside the time and food needed to sustain all these individuals, let’s look at the structure of the tree.

The only way for Cody to be in an infinite joke chain is if at least one of the people he meets with are in an infinite chain as well. In this way, each person in the infinite chain is in the same situation as Cody himself.

Let’s call the probability for a joke to last forever $P_\infty.$

As we reasoned above, the probability that Cody is in an infinite joke chain is the probability that at least one of the people he talks to is. That probability is just the probability that either one of them is $(2pP_\infty)$ minus the probability that they both are $(p^2P_\infty^2)$.