Question

stat 213 assignment 1: problem 3 (1 point) if a and b are two mutually exclusive events with p(a)=0.4 and p(b)=0.5, find the following probabilities: a) p(a∩b)= b) p(a∪b)= c) p(a^c)= d) p(b^c)= e) p((a∪b)^c)= f) p(a∩b^c)=

Explanation:

Step1: Recall definition of mutually - exclusive events

For mutually - exclusive events \(A\) and \(B\), \(P(A\cap B)=0\).

Step2: Use the addition rule for mutually - exclusive events

The formula for \(P(A\cup B)\) when \(A\) and \(B\) are mutually - exclusive is \(P(A\cup B)=P(A)+P(B)\). So \(P(A\cup B)=0.4 + 0.5=0.9\).

Step3: Use the complement rule

The complement rule states that \(P(A^{c})=1 - P(A)\). So \(P(A^{c})=1 - 0.4 = 0.6\).

Step4: Use the complement rule

The complement rule states that \(P(B^{c})=1 - P(B)\). So \(P(B^{c})=1 - 0.5 = 0.5\).

Step5: Use the complement rule

Since \(P(A\cup B)=0.9\), then \(P((A\cup B)^{c})=1 - P(A\cup B)=1 - 0.9 = 0.1\).

Step6: Since \(A\) and \(B\) are mutually - exclusive

\(A\cap B^{c}=A\), so \(P(A\cap B^{c})=P(A)=0.4\).

Answer:

a) \(0\)
b) \(0.9\)
c) \(0.6\)
d) \(0.5\)
e) \(0.1\)
f) \(0.4\)