Check if receiving an academic award and receiving a Fine Arts award are independent of each other.
Dr. G does not need to worry. Receiving the Fine Arts award is independent of receiving the academic award.
Practice makes perfect
To determine of Dr. G has anything to worry about regarding the awards, we should test if receiving an academic award and receiving a Fine Arts award are independent of each other. Recall that events A and B are independent of each other if the probability that both of them occur is the product of the probability of A and the probability of B.
P(A and B) = P(A) * P(B)
To test whether the events are independent in our case, let's first find the probabilities of: receiving the academic award, the Fine Arts award, and both of the awards.
Finding the Probabilities
We can estimate the probability of receiving an award by diving the number of students that received the award by the total number of students. Out of 768 students, 128 won academic awards. Let's find the probability of receiving an academic award.
128/768 = 1/6 ≈ 16.6 %
The probability of receiving an academic award is 16, or about 16.6 %. Next, we know that 48 students won Fine Arts award. Let's divide this number by the total number of students to find the probability of receiving the Fine Arts award.
48/768 = 1/16 = 6.25 %
The probability of receiving the Fine Arts award is 116, or about 6.25 %. Finally, let's find the probability of receiving both awards. We know that 8 out of 768 students received both awards.
8/768 = 1/96 ≈ 1 %
The probability of receiving both awards is 196, or about 1 %.
Independence
Now, let's test if the events of receiving the academic awards and the Fine Arts award are independent. To do so, let's check if the probability of receiving both awards, ( 196), is the product of the probabilities of receiving the academic award, ( 16), and the Fine Arts award, ( 116).
