Worksheet 4: Intro to probability

Worksheet 4: Intro to probability#

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Test your intuition#

  1. Suppose I flip a fair coin, and it comes up heads. If I flip it a second time, is it more likely to come up heads or tails?

  2. Suppose I have a bag with 2 red marbles and 2 blue marbles. I draw a marble from the bag, and it is blue. I don’t return the marble to the bag. If I draw another marble uniformly from the bag, is it more likely to be red or blue?

Coin toss#

I toss a fair coin twice.

  1. What is the probability of the event that the first coin toss is “H”?

  2. What is the probability of the event that I toss at least one “H”?

  3. What is the probability of the event that both coin tosses are “H”?

Marbles#

What is the sample space for the following marbles scenario?

I have a bag with two red marbles and one blue marble. I draw a marble randomly from the bag.

  1. What is the sample space?

  2. Are the outcomes equally likely?

  3. What is the probability of the event that the marble is blue?

Leibniz’s mistake#

  1. True or False: If I roll two six-sided dice, they are equally likely to sum to 11 as they are to sum to 12.

  2. What are the outcomes that give a sum of 11? What are the outcomes that give a sum of 12?

  3. Are the outcomes equally likely?

  4. What is the probability of the event that that the sum of the dice is 12?

  5. What is the probability of the event that that the sum of the dice is 11?

Marbles and Coinflips all the way down#

How can we simulate, with marbles and dice:

  1. A poll in which we ask \(n\) people if they prefer option \(A\) or \(B\)?

  2. An NBA team (say, the Warriors) playing \(5\) games against \(5\) different oponents?

  3. A randomized controlled trial, in which we choose \(m\) sick people at random for the control group, \(m\) sick people at random for the treatment group, treat the treatment group, and observe who gets better?

Axioms of probability#

The following facts are called the axioms of probability.

  1. The probability of any event is always non-negative.

  2. The probability of some outcome happening is \(1\).

  3. For an event \(A\), \(\Pr[A] = 1-\Pr[\bar{A}]\).

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