Question

the probability that a first - year student entering a certain private college needs neither a developmental math course nor a developmental english is 67%. while 26% require a developmental math course and 25% require a developmental english course. find the probability that a first - year student requires both a development math course and a developmental english course.

Explanation:

Step1: Use probability formula

Let \(A\) be the event of needing developmental math and \(B\) be the event of needing developmental English. We know \(P(A) = 0.26\), \(P(B)=0.25\), and \(P((A\cup B)^C)=0.67\). First, find \(P(A\cup B)\) using \(P((A\cup B)^C) = 1 - P(A\cup B)\). So \(P(A\cup B)=1 - 0.67=0.33\).

Step2: Apply inclusion - exclusion principle

The inclusion - exclusion principle states that \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). We want to find \(P(A\cap B)\). Rearranging the formula gives \(P(A\cap B)=P(A)+P(B)-P(A\cup B)\).
Substitute \(P(A) = 0.26\), \(P(B)=0.25\), and \(P(A\cup B)=0.33\) into the formula: \(P(A\cap B)=0.26 + 0.25- 0.33\).

Step3: Calculate the result

\(P(A\cap B)=0.18\)

Answer:

\(0.18\)