Flipping a Coin a Random Number of Times

What this lecture covers

This is a worked recitation problem, not a formal lecture: it applies the ideas of joint PMFs and conditional probability from MIT 6.041 to a two-stage random experiment. A fair four-sided die is rolled to pick a number of coin tosses, and then that many fair coin tosses are performed to count heads.

You'll see how to build a joint probability mass function from a chain of conditional probabilities, read a joint PMF table both ways (fixing the die roll or fixing the coin count), and recover conditional and marginal distributions from that table. It's a good check of whether you can combine the multiplication rule with the binomial distribution.

Key ideas

• Two-stage experiment: the die roll N picks how many times a fair coin is tossed; the coin toss count K then depends on N.
• Law of conditional probability: the joint PMF is built as P(N=n, K=k) = P(K=k | N=n) * P(N=n).
• Conditional independence: once N is known, each of the N coin tosses is conditionally independent, so K given N=n is binomial with parameters n and 1/2.
• Special case N=0: no coin tosses happen, so K is forced to be 0 with probability 1.
• Reading a joint table by row: fixing N=n and dividing by P(N=n) recovers the conditional PMF of K given N.
• Reading a joint table by column: fixing K=k and summing that column gives the marginal P(K=k), needed to get P(N=n | K=k).

Before you watch

• Know the definition of a joint PMF and the multiplication rule for conditional probabilities.
• Be comfortable with the binomial distribution and its formula for a given number of trials and success probability.
• This drill assumes you've already seen discrete random variables and conditioning on an earlier lecture in the course.

Check your understanding

1. Why is K conditionally binomial given N=n, rather than unconditionally binomial?
2. How would the joint PMF table change if the die had a different number of faces?
3. Starting from the joint PMF table, how do you compute P(K=2)?
4. Why must k be no greater than n in every nonzero entry of the joint PMF?