Question

given the following venn diagram: data answer the following questions by filling in each box with the appropriate value. a) ( p(acup b)=) b) ( p(a but not b)=) c) ( p(b but not c)=) d) ( p(neither a nor b nor c)=)

Explanation:

Step1: Calculate \( P(A\cup B) \)

\( P(A\cup B)=P(A)+P(B)-P(A\cap B) \).
\( P(A) = 0.32 + 0.04+0.05 = 0.41 \), \( P(B)=0.04 + 0.26+0.03 + 0.07=0.4 \), \( P(A\cap B)=0.04 + 0.03=0.07 \).
\( P(A\cup B)=0.41 + 0.4-0.07=0.74 \).

Step2: Calculate \( P(A\ but\ not\ B) \)

\( P(A\ but\ not\ B)=P(A)-P(A\cap B) \).
\( P(A) = 0.41 \), \( P(A\cap B)=0.07 \).
\( P(A\ but\ not\ B)=0.41 - 0.07=0.34 \).

Step3: Calculate \( P(B\ but\ not\ C) \)

\( P(B\ but\ not\ C)=P(B)-P(B\cap C) \).
\( P(B) = 0.4 \), \( P(B\cap C)=0.03 + 0.07=0.1 \).
\( P(B\ but\ not\ C)=0.4 - 0.1=0.3 \).

Step4: Calculate \( P(\text{neither }A\text{ nor }B\text{ nor }C) \)

Sum of all given probabilities \(=0.32+0.04 + 0.26+0.05+0.03+0.07+0.14 = 0.91 \).
\( P(\text{neither }A\text{ nor }B\text{ nor }C)=1 - 0.91=0.09 \).

Answer:

a) \( 0.74 \)
b) \( 0.34 \)
c) \( 0.3 \)
d) \( 0.09 \)