Question

7-70. at digital technical institute, the following data was collected:

on academic probation

not on academic probation

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----

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member of honor society

0

30

not a member of honor society

40

170

a. are the events {on academic probation} and {member honor society} associated at this institute?b. are the events {on academic probation} and {member honor society} mutually exclusive at this institute? how many outcomes are in the intersection of the two events?

Explanation:

Step1: Calculate total students

Total = $0 + 40 + 30 + 170 = 240$

Step2: Find P(probation)

$P(\text{probation}) = \frac{0+40}{240} = \frac{40}{240} = \frac{1}{6}$

Step3: Find P(honor society)

$P(\text{honor society}) = \frac{0+30}{240} = \frac{30}{240} = \frac{1}{8}$

Step4: Find P(probation ∩ honor society)

$P(\text{probation} \cap \text{honor society}) = \frac{0}{240} = 0$

Step5: Check association (part a)

If independent, $P(A \cap B) = P(A)P(B)$.
$P(A)P(B) = \frac{1}{6} \times \frac{1}{8} = \frac{1}{48}
eq 0$. So events are associated.

Step6: Check mutual exclusivity (part b)

Mutually exclusive if $P(A \cap B)=0$, which is true here. The intersection has 0 outcomes.

Answer:

a. Yes, the events are associated.
b. Yes, the events are mutually exclusive. The intersection contains 0 outcomes.