Question 48·Hard·Probability and Conditional Probability
A medical lab receives 2,500 blood samples from three clinics. The table shows the percent of all samples that came from each clinic.
| Clinic | Percent of all samples |
|---|---|
| East | 40% |
| North | 35% |
| West | 25% |
That day, of all samples tested positive for a certain marker. Of the samples from the East clinic, 50 tested positive, and of the samples from the West clinic, 50 tested positive.
A sample that tested positive is selected at random from all positive samples received that day. What is the probability that it came from the North clinic?
For conditional probability of the form “from among the positives, what fraction are from North?”, work with counts: (1) find the total number of positives, (2) find North’s number of positives, and (3) compute (North positives) ÷ (total positives), then simplify.
Hints
Convert the overall positive percent to a count
Compute the total number of positive samples using .
Use subtraction to get North positives
If you know the total positives and how many positives came from East and West, you can find North positives by subtraction.
Make the conditional fraction
Your probability is
Desmos Guide
Compute the fraction of positives from North
In Desmos, enter
((0.082*2500)-50-50)/(0.082*2500)
Then use the fraction display to match the result to an answer choice.
Step-by-step Explanation
Compute the total number of positive samples
Since of 2,500 samples tested positive,
So there were 205 positive samples in total.
Find how many positive samples came from North
The positive samples from East and West account for positives.
So the number of positive samples from North is
Form the conditional probability
Selecting from only the positive samples means
Simplify
Simplify:
So the probability is .