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True or False: If A,B and C are three independent events such that P(A)+P(B)=P(C)=p,then\[P(At\;least\;two\;of\;A,B,C\; occur)=3p^2-2p^3\]

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  • (1) Three event A,B,C, associated with a random experiment are independent.
  • $P(A\cap B\cap C)=P(A) P(B) P(C)$
  • (2) Also If A,B,C are independent.
  • $A,B,\bar{C}$ are independent.
  • $\bar{A}, B,C $ are independent.
  • $A, \bar{B},C$ are independent.
P(At least two of A, B, C occures)
=$P(A\cap B\cap \bar{C})+P(A\cap\bar{B}\cap C)+P(\bar{A}\cap B\cap C)+P(A\cap B\cap C)$
Given that $P(A)=P(B)=P(C)$
$P(A\cap B\cap\bar{C})=P(A)P(B)P(\bar{C})$
$P\times P\times (1-P)$
$P(A\cap \bar{B}\cap C)=P(A)P(\bar{B})P(C)$
$P(\bar{A}\cap B\cap C)=P(\bar{A})P(B)P(C)$
=$(1-P)P\times P$
$P(A\cap B\cap C)=P(A)P(B)P(C)$
=$P\times P\times P\times=P^{3}$
P(At least two of A, B, C occures)
Hence given statement is True
answered Jun 13, 2013 by poojasapani_1

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