Probability of two students from overlapping subject groups

Key concept: This is a two-stage without-replacement probability problem built from a Venn diagram. The key is to read each region correctly, then update the sample space after the first draw.

Step by step

Step 1: Read the Venn diagram correctly

The first job is to identify how many students are in each relevant region. The question asks for a student who studies History first, then a second student who studies Geography but not Psychology.

Because the selection is without replacement, the second probability must be based on the reduced group of 99 students. The Venn diagram gives the counts in each subject region, so the answer comes from multiplying the two conditional probabilities.

From the diagram, the number who study History is the total in every region inside the History circle. The number who study Geography but not Psychology is the Geography-only region plus any region that is in Geography and History but not Psychology, depending on how the diagram is arranged. Here the wording “Geography but not Psychology” means we include only students in Geography and exclude any student also in Psychology.

Step 2: Build the two-stage probability

For the first draw:

$P(\text{first is History})=\frac{\text{number studying History}}{100}$

For the second draw, after one History student has been removed, the denominator becomes 99. The numerator is the number of remaining students who study Geography but not Psychology. If the first student came from the History group but not from the Geography-but-not-Psychology group, that count is unchanged.

So the required probability is

$P=\frac{H}{100}\times\frac{G_{\text{not P}}}{99}$

where $H$ is the total number of History students and $G_{\text{not P}}$ is the number of students in Geography only.

Step 3: Common reasoning check

A frequent mistake is to use $\frac{G_{\text{not P}}}{100}$ for the second draw. That would ignore the fact that one student has already been removed. Another mistake is to treat “Geography but not Psychology” as “Geography and not History,” which is not what the question says.

The final answer must come from the exact region counts in the diagram and the multiplication rule for dependent events.

Pitfall alert

A real trap here is misreading the Venn regions under time pressure. Students often count “Geography but not Psychology” as every Geography label they see, even if part of that region overlaps Psychology. Another common error is forgetting that the second draw is from 99 students, not 100. If the first chosen student belongs to the History group, the sample space changes immediately, so the correct method is a conditional probability or a product of two stage probabilities with different denominators.

Try different conditions

If the question changed to: “A student is chosen at random from the 100 students, then another student is chosen from the remaining students. Find the probability that the first student studies Geography and the second student studies History but not Psychology,” the structure would be identical but the region counts would switch. You would first use the total Geography count in the numerator for the first draw, then use the History-only-or-History-without-Psychology count required by the new wording for the second draw. The key skill is translating the phrase into the correct Venn region before multiplying.

Further reading

without replacement, Venn diagram regions, conditional probability