# In a group of 40 students, 22 study Economics, 25 study Law, and 3 study neither of these subjects. Determine the probability that a randomly chosen student studies:

1. Both Economics and Law
2. At Least one of these subjects
3. Economics given that he or she studies Law

I really don't understand how to calculuate the 3rd one.

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## To determine the probability that a randomly chosen student studies Economics and Law, we need to know the number of students who study both subjects. Unfortunately, this information is not provided in the question. Without this information, we cannot calculate the probability.

Moving on to the second question, to calculate the probability that a randomly chosen student studies at least one of these subjects, you need to consider the total number of students who study either Economics or Law or both subjects.

First, we need to find the number of students who study both subjects. However, this information is not given, so we cannot determine that number.

To find the probability that a randomly chosen student studies at least one of these subjects, we can use the principle of inclusion-exclusion. The principle states that:

P(A or B) = P(A) + P(B) - P(A and B)

In this case, A represents the event of studying Economics, and B represents the event of studying Law. P(A) is the probability of studying Economics, P(B) is the probability of studying Law, and P(A and B) is the probability of studying both subjects.

From the given information, we know that 22 students study Economics, 25 students study Law, and 3 students study neither subject. So, to calculate the probability of studying at least one of these subjects, we can use the formula:

P(Economics or Law) = P(Economics) + P(Law) - P(Economics and Law)

P(Economics) = 22/40 (22 students out of 40 study Economics)
P(Law) = 25/40 (25 students out of 40 study Law)

Since we don't have information about the number of students who study both subjects, we cannot determine the probability and provide a precise answer. However, we can calculate an upper bound for the probability of studying at least one of these subjects by assuming that all 25 students who study Law also study Economics. In this case, the probability would be:

P(Economics or Law) = P(Economics) + P(Law) - P(Economics and Law)
= 22/40 + 25/40 - (22 + 3)/40 (assuming all 25 Law students also study Economics)
= 44/40
= 1.1

Therefore, the upper bound for the probability is 1.1. However, please note that this is an unrealistic assumption, and the actual probability could be lower once we have the information about the number of students who study both subjects.

Moving on to the third question, we need to find the probability of studying Economics given that the student studies Law. This probability can be determined using conditional probability.

Let's denote E as the event of studying Economics and L as the event of studying Law.

P(E|L) represents the probability of studying Economics given that the student also studies Law. The conditional probability formula states that:

P(E|L) = P(E and L) / P(L)

To calculate P(E and L), we need to know the number of students who study both subjects - the missing information in the question. Without this information, we cannot calculate P(E and L) and, consequently, cannot find P(E|L).

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b) 21/32
c) 11/20