2  Working with Probabilities

• It is often helpful to think of probabilities as percentages or proportions.
• Furthermore, when working with multiple percentages, it is also helpful to construct hypothetical two-way tables (a.k.a., contingency tables) of counts.
• For the purposes of constructing the table and computing related probabilities, any value can be used for the hypothetical total count.
• When dealing with percentages (or proportions or probabilities) be sure to ask “percent of what?” Thinking in fraction terms, be careful to identify the correct reference group which corresponds to the denominator.

Example 2.1 Are Americans in favor of free tuition at public colleges and universities? According to a study conducted by the Pew Research Center in January 2020

• 83% of Democrats are in favor of free tuition
• 60% of Independents are in favor of free tuition
• 39% of Republicans are in favor of free tuition

Also suppose that¹

• 32% of Americans are Democrats
• 42% of Americans are Independents
• 26% of Americans are Republicans

We’ll use this information to investigate the following questions, as well as a few others.

• What percentage of Americans are in favor of free college tuition?
• What percentage of Americans who are in favor of free college tuition are Democrats?

1. Donny Don’t says the answer to the question “what percentage of Americans who are in favor of free college tuition are Democrats” is 83%. Explain why Donny is wrong without doing any calculations.
   
   
   
   
   

2. For the remaining parts, consider a hypothetical group of 10000 Americans and assume the percentages provided apply to this group. How many people in the group are Democrats?
   
   
   
   
   

3. How many Americans in the group are Democrats who are in favor of free college tuition?
   
   
   
   
   

4. Fill in the counts in each of the cells of the following table.

   	Democrat	Independent	Republican	Total
   In favor of free tuition
   Not in favor of free tuition
   Total				10000

5. What percentage of Americans in this group who are in favor of free college tuition are Democrats? (Answer with both an unreduced fraction and a percent.)
   
   
   
   
   

6. Now answer the original question: What percentage of Americans who are in favor of free college tuition are Democrats? (Hint: did it matter that we used a total of 10000?)
   
   
   
   
   

7. What percentage of Americans who are Democrats are in favor of free college tuition? (Answer with both an unreduced fraction and a percent.)
   
   
   
   
   

8. What percentage of Americans are Democrats in favor of free college tuition? (Answer with both an unreduced fraction and a percent.)
   
   
   
   
   

9. Compare the unreduced fractions for the previous three parts. What is the same? What is different?
   
   
   
   
   

10. What percentage of Americans are in favor of free college tuition?
    
    
    
    
    

• Warning! In general, knowing probabilities of individual events alone is not enough to determine probabilities of combinations of them.

Example 2.2

Suppose that 47% of American adults² have a pet dog and 25% have a pet cat.

1. Donny Don’t says that 72% (which is 47% + 25%) of American adults have a pet dog or a pet cat. Is that necessarily true? If not, is it even possible (in principle anyway) for this to be true? Under what circumstance (however unrealistic) would this be true? Construct a corresponding two-way table.
   
   
   
   
   
2. Given only the information provided, what is the smallest possible percentage of American who adults have a pet dog or a pet cat? Under what circumstance (however unrealistic) would this be true? Construct a corresponding two-way table.
   
   
   
   
   
3. Donny Don’t says that 11.75% (which is 47% \(\times\) 25%) of Americans have both a pet dog and a pet cat. Explain to Donny why that’s not necessarily true. Without further information, what can you say about the percent of American adults who have both a pet dog and a pet cat?
   
   
   
   
   
4. Suppose that 14% of American adults have both a pet dog and a pet cat. What is the percentage of American adults who have a pet dog or a pet cat? Construct a corresponding two-way table. Use your table to show Donny how to correct his error from part 1.
   
   
   
   
   
5. What percentage of American adults who have a pet dog also have a pet cat? Is it 25%?
   
   
   
   
   
6. What percentage of American adults who do not have a pet dog have a pet cat? Is this the same value as in the previous part?
   
   
   
   
   

Example 2.3

A woman’s chances of giving birth to a child with Down syndrome increase with age. The CDC estimates³ that a woman in her mid-to-late 30s has a risk of conceiving a child with Down syndrome of about 1 in 250. A nuchal translucency (NT) scan, which involves a blood draw from the mother and an ultrasound, is often performed around the 13th week of pregnancy to test for the presence of Down syndrome (among other things). If the baby has Down syndrome, the probability that the test is positive is about 0.9. However, when the baby does not have Down syndrome, there is still a probability that the test returns a (false) positive of about⁴ 0.05. Suppose that the NT test for a pregnant woman in her mid-to-late 30s comes back positive for Down syndrome. What is the probability that the baby actually has Down syndrome?

1. Before proceeding, make a guess for the probability in question.

\[ \text{0-20\%} \qquad \text{20-40\%} \qquad \text{40-60\%} \qquad \text{60-80\%} \qquad \text{80-100\%} \]

1. Donny Don’t says: 0.90 and 0.05 should add up to 1, so there must be a typo in the problem. Do you agree?
   
   
   
   
   
2. Construct a hypothetical two-way table of counts to represent the given information.
   
   
   
   
   
3. Use the table to find the probability in question: If NT test for a pregnant woman in her mid-to-late 30s is positive, what is the probability that the baby actually has Down syndrome?
   
   
   
   
   
4. The probability in the previous part might seem very low to you. Explain why the probability is so low.
   
   
   
   
   
5. Compare the probability of having Down Syndrome before and after the positive test. How much more likely is a baby who tests positive to have Down Syndrome than a baby for whom no information about the test is available?
   
   
   
   
   
   

• Remember to ask “percentage of what”? For example, the percentage of babies who have Down syndrome that test positive is a very different quantity than the percentage of babies who test positive that have Down syndrome.
• Probabilities are often conditional on information.
• Conditional probabilities (e.g., probability of Down Syndrome given a positive test) can be highly influenced by the original unconditional probabilities (e.g. probability of Down Syndrome), sometimes called the base rates. Don’t neglect the base rates when evaluating probabilities.
• The example illustrates that when the base rate for a condition is very low and the test for the condition is less than perfect there will be a relatively high probability that a positive test is a false positive.

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1. These values are based on surveys by Gallup, but the values change somewhat over time.

2. These values are based on the 2018 General Social Survey.

3. Source: http://www.cdc.gov/ncbddd/birthdefects/downsyndrome/data.html

4. Estimates of these probabilities vary between different sources. The values in the exercise were based on https://www.ncbi.nlm.nih.gov/pubmed/17350315