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Question
An experiment succeeds twice as often as it fails, what is the probability that in next five trials there will be three successes
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Solution
Success = 2 × fails
p = 2q
⇒ p = 2(1 – p)
p = 2 – 2p
⇒ p + 2p = 2
3p = 2 and p = `2/3`
q = 1 – p = `1 – 2/3`
q = `1/3` and n = 5
The binomial destribution is
P(X = x) = nCxpxqn-x
= `5"C"(2/3)^x (1/3)`
P(three successes) = P(X = 3)
= `5"c"_3 (2/3)^2 (1/3)^(5 - 3)`
= `5"c"_2 (2/3)^3 (1/3)^2` .....`["nc"_x = "nc"_("n" x)]`
= `(5 xx 4)/(1 xx 2) xx 8/27 xx 1/9`
= `10 xx 8/243`
= `80/243`
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