Lab: More Counting

some problems modified from: https://www.probabilitycourse.com/chapter1/1_4_5_solved3.php

Problem 1: Conditional Probability, small game

You are playing a game with a friend. You roll a (6 sided) die and flip a coin. If you roll a 6 or the coin is heads,

you win.

• What is the probability you win?
• What is the probability that you rolled a 6 given that you won?
• What is the probability that the coin was heads given that you won?

Problem 2: Conditional Probability, fair coins

You flip 3 fair coins.

• What is the probability that all 3 are heads?
• What is the probability that 2 are heads?
• What is the probability that at least two are heads?
• What is the probability that 1 is heads?
• What is the probability that at least 1 is heads?
• Given that you have observed at least one heads, what is the probability that there are at least 2 heads?

Problem 3: Conditional Probability, unfair coins

A box has 3 coins. 2 fair, one double sided heads coin.

• You pick a coin at random and toss it. What is the probability that it lands heads up?
• You pick a coin at random and toss it, and get heads. What is the probability that it is the two-headed coin

Problem 4: Family Paradox

https://en.wikipedia.org/wiki/Boy_or_girl_paradox

A family has 2 children (who are either boys or girls). Assume that birth-sex is independent, and equally likely.

• What is the probability that both are girls?
• What is the probability that there is exactly one girl?
• Given that the older child is a girl, what is the probability that both are girls?
• Given that we’ve only met one daughter, what is the probability that both are girls?
• Given that we know that one child is a boy born on Tuesday, what is the probability that the other child is a

• boy?

  • If we pick a 2 child family at random, pick a child at random, see if they are a boy, ask what day he was born, and then figure out the other child.
  • Pick from the universe of 2-children families where one child is a boy born on a Tuesday

Problem 5: Simpson’s Paradox

https://en.wikipedia.org/wiki/Simpson%27s_paradox

You are given the following dataset on a treatment for kidney stones.

Treatment	Stone-size	Total	Successfully Treated
A	small	87	81
A	large	263	192
B	small	270	243
B	large	80	55

Which treatment is more effective is based on success ration (success/total)

• Which procedure is more effective for treating small stones?
• Which procedure is more effective for treating large stones?
• Which procedure is more effective overall?
• Which procedure would you rather have?

Problem 6: Prosecutor’s Fallacy

https://en.wikipedia.org/wiki/Base_rate_fallacy

Imagine that a group of police officers have breathalyzers displaying false drunkenness in 5% of the cases in which

the driver is sober. However, the breathalyzers never fail to detect a truly drunk person. One in a thousand drivers is

driving drunk. Suppose the police officers then stop a driver at random to administer a breathalyzer test. It indicates

that the driver is drunk. No other information is known about them. What is the probability they are drunk?

• Before doing the math (guess/intuition)
• After doing the math