Question

An experiment succeeds twice as often as it fails, what is the probability that in next five trials there will be at least 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(atleast three success)

= P(X ≥ 3)

= P(X = 3) + P(X = 4) + P(X = 5)

= `80/243 + 5"c"_4 (2/3)^4 + (1/3)^(5 - 4) + 5"c"_5 (2/5)^5 (1/3)^(5 - 5)`

= `80/243 + 5"C"_1 (16/81)(1/3) + 5"c"_0 (32/243)(1/3)^0`

= `80/243 + 80/243 + 32/243`

= `(80 + 80 + 32)/243`

= `192/243`