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# Intertwined Conditional Probability | ISI MStat 2016 PSB Problem 4 This is an interesting problem from intertwined conditional probability and Bernoulli random variable mixture, which gives a sweet and sour taste to Problem 4 of ISI MStat 2016 PSB.

## Problem

Let and be three Bernoulli random variables such that and are independent, and are independent, and and are independent.
(a) Show that .
(b) Show that if equality holds in (a), then ### Prerequisites

• Principle of Inclusion and Exclusion • Basic Probability Theory
• Conditional Probability
• iff or or .
• = or; = and

## Solution

#### (a)  We use the fact that and are independent, and are independent, and and are independent. . and be three Bernoulli random variables. Hence, .

#### (b) .

Now, this is just a logical game with conditional probability. . . .  .

Now, is a Bernoulli random variable. So,  . . . . .

Hence, .

This is an interesting problem from intertwined conditional probability and Bernoulli random variable mixture, which gives a sweet and sour taste to Problem 4 of ISI MStat 2016 PSB.

## Problem

Let and be three Bernoulli random variables such that and are independent, and are independent, and and are independent.
(a) Show that .
(b) Show that if equality holds in (a), then ### Prerequisites

• Principle of Inclusion and Exclusion • Basic Probability Theory
• Conditional Probability
• iff or or .
• = or; = and

## Solution

#### (a)  We use the fact that and are independent, and are independent, and and are independent. . and be three Bernoulli random variables. Hence, .

#### (b) .

Now, this is just a logical game with conditional probability. . . .  .

Now, is a Bernoulli random variable. So,  . . . . .

Hence, .

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