Example 2-10: Jury Trial Section
In a jury trial, suppose the probability the defendant is convicted, given guilt, is 0.95, and the probability the defendant is acquitted, given innocence, is 0.95. Suppose that 90% of all defendants truly are guilty. Find the probability the defendant was actually innocent given the defendant is convicted. The video will step you through this example.
Video: Jury Trial Example
Answer
Let Guilty = \(G\)
Innocent = \(I\)
Acquitted = \(A\)
Convicted = \(C\)
\(P(G) = 0.9\) so \(P(I) = 0.1\)
\(P(C | G) = 0.95\) so \(P(A | G) = 0.05\)
\(P(A | I) = 0.95\) so \(P(C | I) = 0.05\)
Need to find: \(P(I | C)\)
\begin{align} P(I\ and\ C) &= P(C | I)*P(I)\\ &= 0.05*0.1\\ &= 0.005 \end{align}
\begin{align} P(C) &= P(G\ and\ C) + P(I\ and\ C)\\ &= (0.95)*(0.9) + ( 0.05)*(0.1)\\ &= 0.855 + 0.005\\ &= 0.86\\ \end{align}
\begin{align} P(I | C) &= \dfrac{P(I\ and\ C) }{P(C)} \\&= \dfrac{0.005}{0.86}\\ &= 0.006 \end{align}
The above example illustrates the use of Bayes' theorem to find "reverse" conditional probabilities.
Suppose we have events $A_1, \dots, A_k$ and event B. If $A_1, \dots, A_k$ are $k$ mutually exclusive events, then...
\( P(A_{i}|B)=\dfrac{P(B | A_{i})P(A_{i})}{\sum_{i} P(B | A_{i})P(A_{i})}=\dfrac{P(B | A_{i})P(A_{i})}{P(B| A_{1})P(A_{1})+P(B |A_{2})P(A_{2})+...+P(B| A_{k})P(A_{k})}\)
Applying this to just two events A and B we have...
\( P(A|B)=\dfrac{P(B | A)P(A)}{ P(B | A)P(A)+P(B| A')P(A')}\)
Example 2-11: Screw Manufacturing Section
A company creates their product using a specially made screw. For financial purposes, the company gets their screws from three different manufacturers. If a screw is defective, it can cause a lot of damage. Here is the table of the proportion of screws from each manufacturer and the probability of obtaining a defective screw.
| Manufacturer | Probability of Company's Screws | Probability of Defective Screw |
|---|---|---|
| A | 0.40 | 0.01 |
| B | 0.25 | 0.02 |
| C | 0.35 | 0.015 |
If a screw is found to be defective, what is the probability that it came from Manufacturer C?
Answer
Let D denote a defective screw. We want \(P(C|D)\). We can use Bayes' Theorem to find this probability.
\begin{align} P(C|D) &=\frac{P(C\cap D)}{P(D)}\\ &=\frac{P(D|C)P(C)}{P(D|C)P(C)+P(D|A)P(A)+P(D|B)P(B)}\\ &=\frac{0.015(0.35)}{0.015(0.35)+0.01(0.40)+0.02(0.25)}\\ &=\frac{0.00525}{0.01425}\\ &=\frac{7}{19}=0.36842 \end{align}
Practical Application: Bayes' Theorem in Diagnostic Testing Section
In diagnostic testing (e.g. drug tests), there are five key concepts:
- Prevalence
-
Prevalence is the probability or proportion of occurrence of a disease or behavior in the population at a particular point in time.
- Example: Proportion of bus drives who use illegal drugs
- Sensitivity and Specificity
-
- Sensitivity is the probability of a positive result given person is actually positive.
- Example: the probability of a home pregnancy test coming up positive for a woman who is actually pregnant
- Specificity is the probability of a negative result given person is actually negative.
- Example: the probability of a home pregnancy test coming up negative for a woman who is not pregnant
- Sensitivity is the probability of a positive result given person is actually positive.
- False Positives and False Negatives
-
- False Positives are when results come back positive for someone who is actually negative
- Example: a home pregnancy test coming up positive for a woman who is not pregnant
- False Negatives are when results come back negative for someone who is actually positive
- Example: a home pregnancy test coming up negative for a woman who is actually pregnant
- False Positives are when results come back positive for someone who is actually negative
Example 2-12: Diabetes Screening Section
Consider the following data on a diabetes screening test based on a non-fasting blood screen test, which is relatively inexpensive and painless.
| Diabetes? | Positive | Negative | Total |
|---|---|---|---|
| Yes | 350 | 150 | 500 |
| No | 1900 | 7600 | 9500 |
| Total | 2250 | 7750 | 10,000 |
Answer
With these results, we see the Sensitivity is 350 out of 500 or 70% and the specificity is 7600 out of 9500 or 80%. The overall prevalence of diabetes is 500 out of 10000 or 5%.
From this test, how many were “missed” (i.e. actually had diabetes – the false negatives) and how many were incorrectly identified as having the disease (i.e. false positives).
The test missed identifying 150 (a false negative rate of 150/500 or 30%) while the false positive rate was 1900/9500 or 20%.
We can see the importance of getting second opinions. What happens with the second opinion (or second test) is that a more expensive and accurate test is used (e.g. a clinical test for pregnancy or a glucose tolerance test for diabetes that requires fasting and a day at a clinic/hospital/doctor’s office). These additional tests are done to verify results before continuing with what can be expensive and uncomfortable treatments.