Conditional probability

The probability of an event given another event has occurred. Typically this will change the sample space as seen below.

Two-way frequency table with experimental data

What is your favorite movie genre?
  Romance Scary other Totals
Front row        
Middle row        
Back row        
Totals        
  1. What is the probability that a student likes scary movies?
  2. What is the probability that a student is from the front row?
  3. What is the probability that a student is from the front row if they like scary movies?

"Effectiveness" of Drug testing

Adapted from https://resources.corwin.com/sites/default/files/lesson_7.1_worksheet_1.pdf

1. What do you know about drug testing?

2. Theoretical situation: Assume that a certain drug test is 98 percent accurate. This means 98 percent of the people who used the drug will test positive and 98 percent of the people who did not use the drug will test negative. Also assume that only 5 percent of people on the job (1 in 20) engage in drug use.

  1. What do the following vocabulary terms mean in regards to test results: true positive, false positive, false negative, true negative?
  2. If a person tests positive, how likely is it that they actually used drugs? (Hint: It may help to assume a large populate is tested, like 100,000 people. Figure out how many people in that population use drugs and how many users/nonusers test positive.)
  3. Do you think such a test should be used?

Further reading:
Liquid Gold: Pain Doctors Soak Up Profits By Screening Urine For Drugs (Schulte & Lucas, 2017)
Racism, Again: Why Drug Tests Are Helping Black Americans Get Jobs Rosen, 2014)


Extra practice and challenges

Two-way frequency table with theoretical data

A rare disease infects 1 out of every 1000 people. A lab test is 99.8% accurate to indicate if a person has the disease or not.

You just received a positive test. What is the probability that you have the disease? [source]

(Hints: Assume 1,000,000 people. Complete the two-way frequency table below.)

  Has disease No disease Totals
Positive test
Negative test
Totals

P(vaccinated | infected) vs. P(infected | vaccinated)


Dice example

What is the probability of rolling a sum of 2 on two six-sided dice?

What is the probability of rolling a sum less than or equal to 5?

What is the probability of rolling a sum of two, given the roll was less than or equal to 5?


Challenge: Second child problem (source)

Mr. Jones has two children, at least one of which is a boy. What is the probability the other child is a boy?

Mr. Smith has two children and the eldest is a boy. What is the probability that both children are boys?


Super Challenge: Monty Hall problem (!)

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?

See Cut-the-knot for more information


Resources