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  • State True or False for the statements in the Exercise. If A, B and C are three independent events such that P(A) = P(B) = P(C) = p, then P (At least two of A, B, C occur) = 3p^2 – 2p^3

State True or False for the statements in the Exercise.

If A, B and C are three independent events such that P(A) = P(B) = P(C) = p, then P (At least two of A, B, C occur) =3p^2 - 2p^3

Answers (1)

Solution

Let A, B, C be the occurrence of events A, B and C and A’, B’ and C’ not occurrence.
 P(A) = P(B) = P(C) = p and P(A’) = P(B’) = P(C’) = 1-p
P (At least two of A, B, C occur) = P(A ∩ B ∩ C’) + P(A ∩ B’ ∩ C) + P(A’ ∩ B ∩ C) + P(A ∩ B ∩ C)
events are independent:

\\\mathrm{P} (Atleast \ two \ of \ \mathrm{A}, \mathrm{B}, C \ occur )=\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B}) \mathrm{P}\left(\mathrm{C}^{\prime}\right)+\mathrm{P}(\mathrm{A}) \mathrm{P}\left(\mathrm{B}^{\prime}\right) \mathrm{P}(\mathrm{C})+\mathrm{P}\left(\mathrm{A}^{\prime}\right) \mathrm{P}(\mathrm{B}) \mathrm{P}(\mathrm{C})+\mathrm{P}\left(\mathrm{A}\right) \mathrm{P}(\mathrm{B}) \mathrm{P}(\mathrm{C})\\=3 \mathrm{p}^{2}(1-\mathrm{p})+p^3
=3 p^{2}-2p^{3}
Hence, the statement is true.

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