Q. 255.0( 1 Vote )

# Let A and B be two independent events such that P(A) = p_{1} and P(B) = p_{2}. Describe in words the events whose probabilities are:

i. p_{1}p_{2}

ii. (1–p_{1}) p_{2}

iii. 1–(1–p1)(1–p2)

iv. p_{1}+p_{2} = 2p_{1}p_{2}

Answer :

If the events are said to be independent, if the occurrence or non occurrence of one does not affect the probability of the occurrence or non occurrence of other.

_{i.} p_{1}p_{2}

p_{1}p_{2}=P(A)P(B)

Both events A and B will occur

_{ii.} (1–p_{1}) p_{2}

(1–p_{1}) p_{2}=[1–P(A)]P(B)

Event A does not occur but event B occur.

iii. 1–(1–p1)(1–p2)

1–(1–p1)(1–p2)=[1–(1–P(A))(1–P(B))]

At least one of the event will occur

iv. p_{1}+p_{2} = 2p_{1}p_{2}

p_{1}+p_{2} = 2p_{1}p_{2}

P(A)+P(B)=2P(A)P(B)

P(A)+P(B)–2P(A)P(B)=0

P(A)–P(A)P(B)+P(B)–P(A)P(B)=0

P(A)[1–P(B)]+P(B)[1–P(A)]=0

Exactly one of A and B occurs

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