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.
there are 47 places in the two classes that are filled by 37 students
... 3 of the 40 students are in neither class
if B is the students in both classes
... 22 + 25 - B = 37
1. 10 of 40
2. 37 of 40 students in at least one class
3. 25 in Law ... 10 in both ... 10 of 25
Oh, probability can be a tricky subject sometimes, but never fear, Clown Bot is here to help! Let's break it down, step by step:
1. To find the probability that a student studies both Economics and Law, we need to count how many students study both and divide it by the total number of students. But wait! We don't have that information. So, let's assume that the number of students studying both is "x." Now, we know that there are 22 students studying Economics and 25 studying Law. Since we're assuming "x" students are studying both, we subtract that from both numbers: 22 - x students studying only Economics and 25 - x students studying only Law.
Since the total number of students is 40, we can set up an equation: (22 - x) + (25 - x) + x + 3 = 40. Let's solve that equation:
22 - x + 25 - x + x + 3 = 40
50 - x = 40
-x = -10
x = 10
So, we have 10 students studying both Economics and Law. The probability is then 10/40, which simplifies to 1/4 or 25%.
2. To find the probability that a student studies at least one of these subjects, we need to count the total number of students studying Economics, Law, or both, and divide it by the total number of students. From the previous calculation, we found that 10 students study both subjects. So, let's add up the students studying only Economics (22 - 10 = 12), the students studying only Law (25 - 10 = 15), and the students studying both (10). The total is 12 + 15 + 10 = 37. Therefore, the probability is 37/40, which simplifies to 37.5%.
3. Now comes the tricky one! We want to find the probability of a student studying Economics given that he or she studies Law. This is called a conditional probability. To calculate it, we need to divide the number of students studying Economics and Law by the number of students studying Law. From the previous calculation, we found that 10 students study both Economics and Law. And from the given information, 25 students study Law. So, the probability is 10/25, which simplifies to 2/5 or 40%.
Remember, Clown Bot is here to bring a smile to your face while helping you with probabilities!
To determine the probability that a randomly chosen student studies both Economics and Law, we can use the formula:
P(Economics and Law) = P(Economics) + P(Law) - P(Economics or Law)
Given:
Number of students studying Economics (E) = 22
Number of students studying Law (L) = 25
Number of students studying neither subject = 3
Total number of students (T) = 40
Let's calculate the probabilities one by one:
1. Probability of studying both Economics and Law (Economics and Law):
We need to find the number of students who study both subjects. Since we don't have this information, we can assume that the number of students studying both subjects is the minimum of the number studying Economics and Law.
Number of students studying both subjects = min(E, L) = min(22, 25) = 22
Probability of studying both Economics and Law:
P(Economics and Law) = Number of students studying both subjects / Total number of students
P(Economics and Law) = 22 / 40
P(Economics and Law) = 11/20
2. Probability of studying at least one of these subjects:
This means that a student studies either Economics or Law or both.
Probability of studying at least one of these subjects:
P(Economics or Law) = 1 - P(neither Economics nor Law)
Number of students studying neither subject = 3
Probability of not studying either Economics or Law:
P(neither Economics nor Law) = Number of students studying neither subject / Total number of students
P(neither Economics nor Law) = 3 / 40
Probability of studying at least one of these subjects:
P(Economics or Law) = 1 - P(neither Economics nor Law)
P(Economics or Law) = 1 - (3 / 40)
P(Economics or Law) = 37 / 40
3. Probability of studying Economics given that the student studies Law:
To calculate this probability, we need to find the number of students who study both Economics and Law and divide it by the total number of students studying Law.
Number of students studying both subjects = 22 (as calculated earlier)
Number of students studying Law = 25 (given)
Probability of studying Economics given that Law is studied:
P(Economics | Law) = Number of students studying both subjects / Total number of students studying Law
P(Economics | Law) = 22 / 25
To calculate the probability that a randomly chosen student studies Economics and Law, we can use the principle of inclusion-exclusion.
1. Both Economics and Law: In this case, we need to find the number of students who study both Economics and Law. The total number of students who studied Economics is 22, and the total number of students who studied Law is 25. However, we need to subtract the 3 students who study neither of these subjects. So, the number of students who study both Economics and Law is 22 + 25 - 3 = 44. Therefore, the probability is 44/40, or 11/10.
2. At least one of these subjects: To find the probability that a student studies at least one of the subjects, we need to find the number of students who study Economics or Law or both. The total number of students is 40, and the number of students who study neither subject is 3. So, the number of students who study at least one of these subjects is 40 - 3 = 37. Therefore, the probability is 37/40.
3. Economics given that the student studies Law: To find this probability, we need to find the number of students who study both Economics and Law and divide it by the number of students who study Law. From question 1, we know that the number of students who study both Economics and Law is 44. The number of students who study Law is 25. Therefore, the probability is 44/25.