Question

the prerequisite for a required course is that students must have taken course a or course b. by the time they are juniors, 50% of the students have taken course a, 26% have taken course b, and 9% have done both.
a) what percent of the juniors are ineligible for the course?
b) whats the probability that a junior who has taken course a has also taken course b?
c) are taking the two courses disjoint events? explain
d) are taking the two courses independent events? explain.
hint: a venn diagram will help here

Explanation:

Step1: Calculate the percentage of eligible students (a)

Use the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).
Let \(P(A) = 56\%\), \(P(B)=26\%\), \(P(A\cap B)=9\%\).
\(P(A\cup B)=56\% + 26\% - 9\%=73\%\)

Step2: Calculate the conditional probability (b)

The formula for conditional probability is \(P(B|A)=\frac{P(A\cap B)}{P(A)}\).
Substitute \(P(A\cap B) = 9\%\) and \(P(A)=56\%\).
\(P(B|A)=\frac{9\%}{56\%}\approx 16.07\%\)

Step3: Check for disjoint events (c)

Disjoint events satisfy \(P(A\cap B) = 0\).
Since \(P(A\cap B)=9\%
eq0\), the events are not disjoint.

Step4: Check for independent events (d)

Independent events satisfy \(P(A\cap B)=P(A)\times P(B)\).
\(P(A)\times P(B)=56\%\times26\% = 14.56\%
eq9\%=P(A\cap B)\), so the events are not independent.

Answer:

a) \(73\%\)
b) Approximately \(16.07\%\)
c) No, because \(P(A\cap B)
eq0\)
d) No, because \(P(A\cap B)
eq P(A)\times P(B)\)