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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.

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

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

We use the fact that and are independent, and are independent, and and are independent.

.

and be three Bernoulli random variables. Hence,

.

.

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.

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

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

We use the fact that and are independent, and are independent, and and are independent.

.

and be three Bernoulli random variables. Hence,

.

.

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

.

.

.

.

Now, is a Bernoulli random variable. So, .

.

.

.

.

Hence,

.

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