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Probability · Lecture 21 of 76 · 8:43

Flipping a Coin a Random Number of Times

Flipping a Coin a Random Number of Times on YouTube

Study guide

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?

Vocabulary

two-stage experiment (noun)
An experiment where the result of one random step determines the setup of the next.
Rolling a die to pick the number of coin tosses is a two-stage experiment.
conditional independence (noun)
Independence between events that only holds once you know a third event's value.
Given N, each coin toss is conditionally independent.
binomial (adjective)
Relating to a fixed number of independent yes/no trials with the same success probability.
K is binomial given N equals n.
special case (noun)
A particular instance of a general rule that needs separate attention.
N equals 0 is a special case where K must be 0.
fair (die) (adjective)
Having every face equally likely to come up.
A fair four-sided die is rolled to pick the number of tosses.
toss (a coin) (verb)
To flip a coin to get a random heads or tails result.
The coin is tossed N times, where N is decided by the die.
row (of a table) (noun)
A horizontal line of values in a table.
Fixing N and reading across the row gives the conditional PMF.
column (of a table) (noun)
A vertical line of values in a table.
Summing a column gives the marginal probability.
force (a value) (verb)
To make a value certain, with no other possibility.
N equals 0 forces K to be 0 with probability 1.
recover (a distribution) (verb)
To find one distribution starting from a related, larger one.
You can recover the marginal distribution from the joint table.
nonzero entry (noun)
A table value that is not equal to zero.
Every nonzero entry requires k to be no greater than n.
worked drill (noun)
A fully solved practice problem meant to build a specific skill.
This is a worked drill on joint and conditional PMFs.
check (understanding) (verb)
To confirm that a concept has been correctly understood.
This drill helps check whether you can combine two probability rules.
depend on (phrase)
To be affected by or determined by something else.
The coin toss count K depends on the die roll N.
four-sided die (noun)
A die shaped so it can only land on one of four faces.
A fair four-sided die decides how many coin tosses happen.
read (a table) (verb)
To interpret values correctly from a table by row or column.
You can read the joint PMF table either by row or by column.
not a formal lecture (phrase)
Describing a session that is a practice exercise rather than new taught material.
This is a worked recitation problem, not a formal lecture.
chain (of conditioning) (noun)
A linked sequence of conditional probability steps.
The joint PMF is built as a chain of conditioning steps.
good check (noun)
A useful test of whether a skill has really been understood.
This is a good check of whether you can combine two rules together.
apply (ideas) (verb)
To use a known concept to solve a new problem.
This drill applies the ideas of joint PMFs to a two-stage experiment.

Chapters

From the YouTube description

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
View the complete course: http://ocw.mit.edu/6-041SCF13
Instructor: Kuang Xu

License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu

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