Question

Conditional Probability

Activity 1: CHC Student Survey

Suppose a survey of 100 randomly selected CHC students resulted in a sample of 60 male and 40 female students. Of the males, 2/3 graduated from a high school in Philadelphia, while the remainder had high school diplomas from out-of-the-city. Of the females, 3/4 were from Philadelphia high schools. This information is represented in the following contingency table:

	Phila. HS	Out-of-City	Totals
Male	40	20	60
Female	30	10	40
Totals	70	30	100

1. If I randomly select a student from the survey, what is the probability that the student is a Philadelphia high school graduate, P(C)?
2. What is the probability that the student is female, P(F)?
3. What is the probability that the student both graduated from high school in Philadelphia and is female, P(C and F)?
4. What is the probability that the student graduated from high school in Philadelphia, given that the student is female, P(C | F)?
5. What is the probability that the student is female, given that the student graduated from high school in Philadelphia, P(F | C)?

          

MATH 227 – Introduction to Probability and Statistics

Module 5 Homework                                    Name: ______________________________

Activity 2: Coronavirus Testing

Assume there is a test for the SARS-CoV-2 virus (the virus that causes COVID-19) that is 98% accurate; i.e. if someone has the SARS-CoV-2 virus the test will be positive 98% of the time, and if one does not have it, the test will be negative 98% of the time. Assume further that 0.5% of the population actually has the SARS-CoV-2 virus. Imagine that you have taken this test and your doctor somberly informs you that you’ve tested positive. How concerned should you be?

To answer this, let’s make a table for a hypothetical town.

• Suppose the population of the town is 10,000.
• Calculate how many have the virus (0.5% of the population) and how many do not (99.5% of the population). Write your figures in the table below.
• Determine how many will test positive and how may will test negative (98% accurate). Do this separately for the “Has HIV” group and the “No HIV” group. Write your figures in the table below.

	Has Virus (0.5%)	No Virus (99.5%)	Totals
Positive
Negative
Totals			10,000

1. Based on the figures from your table, calculate the probability that the test is positive given that the person has the virus, P(Pos | V). Does your calculation agree with the 98% accuracy rate? If not, go back and check your calculations.
2. Now calculate the probability that a person has the virus given that the test was positive, P(V | Pos).
3. How does the result from part (b) compare with what you expected? Would this increase or decrease your worry if you got a positive test?
4. Can you explain why the result was different than what you expected?

Answer 1

Activity 1:

probability that the student is a Philadelphia high school graduate, P(C) =

70/100

= 0.7

probability that the student both graduated from high school in Philadelphia and is female, P(C and F) =

= 30/100

= 0.30

probability that the student graduated from high school in Philadelphia, given that the student is female, P(C | F)=

= 30/40

= 0.75

probability that the student is female, given that the student graduated from high school in Philadelphia, P(F | C)=

= 30/70

= 0.4286

Thanks in advance!

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