240 people are taking science classes. There are 104 people studying chemistry, 132 people studying biology, and 68 of those are studying both. What is the probability that a person is studying chemistry given that they are studying biology?
The key word in this question is “given,” which tells us that we’re looking for a conditional probability, the conditional probability that somebody is studying chemistry with the knowledge that they are studying biology. Conditional probabilities can be expressed using the notation of a vertical line. We write the probability of 𝐶, chemistry, given 𝐵, biology.
Now, to help us answer this question, let’s organize the information we’ve been given using a Venn diagram. We have two overlapping circles on our Venn diagram, one to include the students who are studying chemistry and one to include the students who are studying biology. The overlap of our Venn diagram represents the students who are studying both chemistry and biology, whereas the region inside the Venn diagram but outside the circles represents students who are studying neither chemistry nor biology. Perhaps these are people who are studying physics.
Now, let’s add the information given in the question onto our Venn diagram. We’re told that there are 104 people studying chemistry, 132 people studying biology, but the key bit is that 68 of these people are studying both. This means that the number in the centre of our diagram, the overlap or intersection of chemistry and biology, is 68.
We also know that there are 104 people studying chemistry. But this 104 is all the people studying chemistry. It includes the 68 people who are also studying biology. So the number of people who are studying chemistry only is 104 minus 68. That’s 36. We know that there are 132 people studying biology. But again this 132 includes the 68 people who are also studying chemistry. So the number of people studying biology only is 132 minus 68. That’s 64.
Finally, the number outside the circles but still inside the Venn diagram, this is the people who are studying neither chemistry nor biology. We find this by subtracting the three values we’ve already calculated from the total of 240. And it gives 72.
So now that we’ve got our Venn diagram, we can work out this probability. And we’re going to look at two methods. The first is just a logical approach. Remember, we’re asked for the probability that somebody is studying chemistry given that they are studying biology, which means we already know that this person studies biology. This means that we’re no longer looking at the full group of 240 people. Instead, we’re just looking at the subset of this group who study biology. That’s all the people in the pink circle. The number of people in the circle is 68 plus 64. It’s 132. The 132 people we were told in the question study biology. So we’re choosing one of these 132 people, which means that our probability will be a fraction with a denominator of 132.
For the numerator, we need the number of people who also study chemistry. That’s all of the people who are in the overlap of the two circles. It’s 68. So if we know that the person we’ve chosen studies biology, this means there are 132 possible people. And we know that 68 of them also study chemistry. So the probability that a person studies chemistry given that they are studying biology is 68 over 132. This fraction can and should be simplified by dividing both the numerator and denominator by four to give the simplified fraction 17 over 33.
Now, let’s look at a second method, a slightly more formal method, in which we’re going to use the formula for conditional probability. The conditional probability formula tells us that the probability of an event 𝐴 occurring, given that we know event 𝐵 has already occurred, is equal to the probability of the intersection of 𝐴 and 𝐵. That’s the probability that 𝐴 and 𝐵 both occur divided by the probability of 𝐵.
If we allow the letter 𝐵 here to represent biology and just swap the 𝐴s for 𝐶s to represent chemistry, then we see that the probability that someone studies chemistry given that they study biology is equal to the probability that they study both chemistry and biology divided by the probability they study biology. We can use either our Venn diagram or the information given in the question to work out each of these probabilities.
Remember, there are 240 people in total. So each probability will be a fraction with a denominator of 240. The number of people who study both chemistry and biology is 68. So the probability of the intersection of 𝐶 and 𝐵 is 68 over 240. The number of people who study biology, regardless of whether or not they also study chemistry, is 132. So the probability that somebody studies biology is 132 over 240.
Now we can substitute these probabilities into our conditional probability formula. And to avoid stacking fractions on top of each other, we’ll write this as a division, 68 over 240 divided by 132 over 240. But remember, if we want to divide by a fraction, then we need to flip or invert that fraction and, instead of dividing, we multiply. So our calculation becomes 68 over 240 multiplied by 240 over 132.
Now, before we attempt this multiplication, we can do some cross-cancellation. There’s a factor of 240 in the denominator of the first fraction and in the numerator of the second. So they’ll cancel out to one. If we perform the multiplication, we get 68 multiplied by one over one multiplied by 132. That’s 68 over 132. Notice that this is the same as the unsimplified probability we found using our first method. So the fraction will cancel down in the same way to give 17 over 33.
So we’ve used two methods to show that the conditional probability that somebody is studying chemistry given that they are studying biology is 17 over 33. This fraction can’t be simplified any further as the numerator and denominator have no common factors other than one.