# Probability-Elements of Measure Theory (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 3 of 3

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## Question number: 1

» Probability » Elements of Measure Theory

Appeared in Year: 2013

### Describe in Detail

If the probability for n independent events are p _{1}, p _{2}, …, p _{n}, then prove that:

(i) none of the events will occur

(ii) at least one event will occur

(iii) at most one event will occur;

### Explanation

Let A _{1 }, A _{2}, …, A _{n}, n independent events with probabilities p _{1}, p _{2}, …, p _{n}.

(i) Find the probability that none of the events will occur

Since events are independent and the probability that one event does not occur

## Question number: 2

» Probability » Elements of Measure Theory

Appeared in Year: 2014

### Describe in Detail

Suppose that all the four outcomes 0 _{1}, 0 _{2}, 0 _{3} and 0 _{4} of an experiment are equally likely. Define A = (0 _{1}, 0 _{4}), B = (0 _{2}, 0 _{4}) and C = (0 _{3}, 0 _{4}). What can you say about the pairwise independence and mutually independence of the events A, B and C?

### Explanation

Given that there are four outcomes. Define A = (O _{1}, O _{4}), B = (O _{2}, O _{4}), C = (O _{3}, O _{4})

Their intersection is and Union is AUBUC = (O _{1}, O _{2}, O _{3}, O

## Question number: 3

» Probability » Elements of Measure Theory

Appeared in Year: 2012

### Describe in Detail

Of three independent events A, Band C, A only happens with probability ¼, B only happens with probability 1/8 and C only happens with probability 1/12. Find the probability that at least one of these three events happens.

### Explanation

Given that P (A) =1/4, P (B) =1/8, P (C) =1/12

then probability that at least one event of these three events happens is

The events are independent because only one event happens