Question

An experiment succeeds twice as often as it fails, what is the probability that in next five trials there will be three successes

Answer 1

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) = nC_xp^xq^n-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`