Let $E_{1}$ and $E_{2}$ be two independent events such that $P (E_{1}) = P_{1}$ and $P (E_{2}) = P_{2}$ . Describe in words of the events whose probabilities are

(i) $P_{1} P_{2}$

(ii) $(1 - P_{1}) P_{2}$

(iii) $1 - (1 - P_{1}) (1 - P_{2})$

(iv) $P_{1} + P_{2} - 2 P_{1} P_{2}$

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Solution

$P (E_{1}) = P_{1}$ and $P (E_{2}) = P_{2}$

(i) $P_{1} P_{2} \Rightarrow P (E_{1}) . P (E_{2}) = P (E_{1} \cap E_{2})$

So, $E_{1}$ and $E_{2}$ occur.

(ii) $(1 - P_{1}) P_{2} = P (E_{1})^{'} . P (E_{2}) = P (E_{1}^{'} \cap E_{2})$
So, $E_{1}$ does not occur but $E_{2}$ occurs.

(iii) $1 - (1 - P_{1}) (1 - P_{2}) = 1 - P (E_{1})^{'} P (E_{2})^{'} = 1 - P (E_{1}^{'} \cap E_{2}^{'}) = 1 - [1 - P (E_{1} \cup E_{2})] = P (E_{1} \cup E_{2})$
So, either $E_{1}$ or $E_{2}$ or both $E_{1}$ and $E_{2}$ occurs.

(iv) $P_{1} + P_{2} - 2 P_{1} P_{2} = P (E_{1}) + P (E_{2}) - 2 P (E_{1}) . P (E_{2}) = P (E_{1}) + P (E_{2}) - 2 P (E_{1} \cap E_{2}) = P (E_{1} \cup E_{2}) - P (E_{1} \cap E_{2})$
So, either $E_{1}$ or $E_{2}$ occurs but not both.

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Let $E_{1}$ and $E_{2}$ be two independent events such that $P (E_{1}) = P_{1}$ and $P (E_{2}) = P_{2}$ . Describe in words of the events whose probabilities are

$1 - (1 - P_{1}) (1 - P_{2})$

E_{1} a n d E_{2}

are events in a probability space satisfying the following constraints

P (E_{1}) = P (E_{2}); P (E_{1} \cup E_{2}) = 1; E_{1} a n d E_{2}

are independent then

P (E_{1}) =

E_{1}

and

E_{2}

are two events such that

P (E_{1}) = \frac{1}{4}

P (\frac{E_{2}}{E_{1}}) = \frac{1}{2}

and

P (\frac{E_{1}}{E_{2}}) = \frac{1}{4}

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Q. If

E_{1}

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E_{2}

are two events such that

P (E_{1}) = \frac{1}{4}, P (\frac{E_{2}}{E_{1}}) = \frac{1}{2}

and

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then which of the following is/are correct?

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Let E1 and E2 be two independent events such that P(E1)=P1 and P(E2)=P2. Describe in words of the events whose probabilities are (i) P1P2 (ii) (1−P1)P2 (iii) 1−(1−P1)(1−P2) (iv) P1+P2−2P1P2

Let $E_{1}$ and $E_{2}$ be two independent events such that $P (E_{1}) = P_{1}$ and $P (E_{2}) = P_{2}$ . Describe in words of the events whose probabilities are

(i) $P_{1} P_{2}$

(ii) $(1 - P_{1}) P_{2}$

(iii) $1 - (1 - P_{1}) (1 - P_{2})$

(iv) $P_{1} + P_{2} - 2 P_{1} P_{2}$