An experiment succeeds twice as often as it fails. Then find the probability that in the next 6 trials, there will be at least 4 sucesses.

Answer

N/a

Explanation

Let the probability of success is `p` and the probability of failure is `q`, then
`p+q=1` (from probability theorem)
and `p=2q` (given)
`implies2q+q=1impliesq=1/3`
`:.p=1-1/3=2/3`
Let `X` be the random variable which represents the number of successes in 6 trials.
Using binomial distribution
`P(X=r)=.^(n)C_(r)p^(r)q^(n-r)=.^(6)C_(r)(2/3)^(r)(1/3)^(6-r)`
`P` (at least 4 sucesses)
`P(Xge4)`
`=P(4)+P(5)+P(6)`
`=.^(6)C_(r)p^(4)q^(2)+.^(6)C_(5)p^(5)q^(1)+.^(6)C_(6)p^(6)q^(0)`
`=p^(4)(.^(6)C_(2)q^(2)+.^(6)C_(1)pq+.^(6)C_(0)p^(2))`
`=(2/3)^(4)[(6xx5)/(1xx2)(1/3)^(2)+6/1 . 2/3 . 1/3+(2/3)^(2)]`
`=(2/3)^(4)[15/9+12/9+4/9]=31/9(2/3)^(4)`