A university has 1500 students out of which 500 are history students, 800 are English students and 200 are both history and English students. Are the events of a student being an English student or a history student independent ?
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To determine whether the events of being an English student and being a history student are independent, we need to compare the probabilities of these events occurring together versus occurring separately.
Let's calculate the probabilities:
1. Probability of being an English student (E):
P(E) = Number of English students / Total number of students
= 800 / 1500
= 0.5333
2. Probability of being a history student (H):
P(H) = Number of history students / Total number of students
= 500 / 1500
= 0.3333
3. Probability of being an English student and a history student (E ∩ H):
P(E ∩ H) = Number of students who are both English and history students / Total number of students
= 200 / 1500
= 0.1333
Now, if the events of being an English student and being a history student are independent, then the probability of both events occurring should be equal to the product of individual probabilities.
P(E ∩ H) = P(E) * P(H)
0.1333 = 0.5333 * 0.3333
0.1333 ≠ 0.1778
Since the probability of both events occurring (0.1333) is not equal to the product of individual probabilities (0.1778), we can conclude that the events of a student being an English student and a history student are dependent.