JIPMAT 2022 (QA) - Given below are two statement based on the following If A and B are independent events such that P(A)=p, P(B)=2 p and P (exactly one of A, B)=5/9 Statement I: p=1/3 Statement II: p=5/12 In the light of the above statements, choose the correct answer form the question given below | PYQs + Solutions

JIPMAT 2022

Modern Math

Probability

Medium

Given below are two statement based on the following
If $A$ and $B$ are independent events such that $P(A)=p, P(B)=2 p$ and $P$ (exactly one of $A, B)=\frac{5}{9}$
Statement I: $\mathrm{p}=\frac{1}{3}$
Statement II: $\mathrm{p}=\frac{5}{12}$
In the light of the above statements, choose the correct answer form the question given below

Both Statement I and Statement II are true

Both Statement I and Statement II are false

Statement I is true but Statement II is false

Statement I is false but Statement II is true

Correct Option: 1

For independent events:

\newline

Given

P(A) = p

P(B) = 2p

P(\text{exactly one}) = \frac{5}{9}

P(\text{exactly one}) = P(A\bar{B} \cup \bar{A}B)

= P(A)(1-P(B)) + P(B)(1-P(A))

Therefore:

\frac{5}{9} = p(1-2p) + 2p(1-p)

= p - 2p^2 + 2p - 2p^2

\newline

= 3p - 4p^2

\newline

\Rightarrow 4p^2 - 3p + \frac{5}{9} = 0

\newline

\Rightarrow 36p^2 - 27p + 5 = 0

For

p = \frac{1}{3}

(Statement I):

\newline

\Rightarrow 36(\frac{1}{9}) - 27(\frac{1}{3}) + 5 = 0

✓

For

p = \frac{5}{12}

(Statement II):

\newline

\Rightarrow 36(\frac{25}{144}) - 27(\frac{5}{12}) + 5 = 0

✓

\newline

Therefore, both statements are true.