Let A and B be two independent events such that P( A )=0.7, P( B )=k, P(A B)=0.8 What is the value of k ?

Correct answer: C. (2)/(7) P( A )=0.7 P(A)=0.3P( B )=k P(B)=1-kP(A B)=0.8 For independent events P(A B)=P(A) · P(B)P(A B)=P(A)+P(B)-P(A) · P(B) 0.8=0.3+1-k-0.3(1-k) 0.5=1-k-0.3+0.3 k 0.5=0.7-0.7 k 0.7 k=0.2 k=(2)/(7)

NDA 2023

Exam: NDA
Year: 2023
Subject: Mathematics
Chapter: Probability

NDA 2023 · Mathematics · Probability

Let A and B be two independent events such that P( A )=0.7, P( B...

Let

A

and

B

be two independent events such that

P(\bar{A})=0.7, P(\bar{B})=k, P(A \cup B)=0.8

What is the value of k ?

Options

A $\frac{5}{7}$
B $0$
C $\frac{2}{7}$
D $\frac{1}{7}$

Correct answer

C. $\frac{2}{7}$

Step-by-step solution

P(\bar{A})=0.7 \Rightarrow P(A)=0.3P(\bar{B})=k \Rightarrow P(B)=1-kP(A \cup B)=0.8

For independent events

P(A \cap B)=P(A) \cdot P(B)P(A \cup B)=P(A)+P(B)-P(A) \cdot P(B)\Rightarrow \quad 0.8=0.3+1-k-0.3(1-k)\Rightarrow \quad 0.5=1-k-0.3+0.3 k\Rightarrow \quad 0.5=0.7-0.7 k\Rightarrow \quad 0.7 k=0.2 \Rightarrow k=\frac{2}{7}

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