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Probability

Short Answer Type

771. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that at least one is a girl?

Let the first child be denoted by capital letter and the second i.e. younger one by a smaller letter.
∴ S = {Bb, Bg, Gb, Gg}
Let E : both children are girls
∴ E = {Gg}
Let F : at least one is a girl
$therefore space space space space space space space space straight F space equals space open curly brackets Bg comma space Gb comma space Gg close curly brackets therefore space space space space straight E space intersection space straight F space equals space open curly brackets Gg close curly brackets therefore space space space straight P left parenthesis straight F right parenthesis space equals space 3 over 4 comma space space space straight P left parenthesis straight E intersection straight F right parenthesis space equals space 1 fourth$
Required probability = $straight P left parenthesis straight E space left enclose straight F right parenthesis space equals space space fraction numerator straight P left parenthesis straight E intersection straight F right parenthesis over denominator straight P left parenthesis straight F right parenthesis end fraction space equals fraction numerator begin display style 1 fourth end style over denominator begin display style 3 over 4 end style end fraction space equals 1 third$

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772.

A family has two children. What is the probability that both the children are boys given that at least one of them is a boy?

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773. Mother, father and son line up at random for a family picture
E : son on one end, F : father in middle, Determine P(E | F).

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774.

A couple has two children,
Find the probability that both children are males, if it is known that at least one of the children is male.

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775.

A couple has two children,
Find the probability that both children are females, if it is known that the elder child is a female.

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Long Answer Type

776.

An electronic assembly consists of two subsystems, say A and B. From the previous testing procedures, the following probabilities are assumed to be known:
P( A fails) = 0.2
P(B fails alone) = 0.15
P(A and B fail) = 0.15
Evaluate the following probabilities:
(i) P( A fails $left enclose straight B space has space failed end enclose$ ) (ii) P(A fails alone)

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777. Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’.

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778. Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it shows tail, then throw a die. Find the conditional probability of the event that ‘the die shows a number greater than 4’ given that ‘there is at least one tail.’

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Multiple Choice Questions

779.

If $straight P left parenthesis straight A right parenthesis space equals space 1 half comma space space space straight P left parenthesis straight B right parenthesis space equals space 0 comma space space then space straight P left parenthesis straight A space left enclose straight B right parenthesis$ is

0
$1 half$
not defined
not defined

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780. If A and B are events such that P(A|B) = P(B|A), then

A ⊂ B but A ≠ B
A = B
A ∩ B = ϕ
A ∩ B = ϕ

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