Question

An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.

Answer 1

The probability of success is twice the probability of the failure.

Let the probability of failure be x.

Then the probability of success is given by 2x.

According to the given condition, the sum of both the probabilities is equal to 1.

x+2x=1 3x=1 x= 1 3

Let probability of success be p and probability of failure be q.

p= 1 3 q= 2 3

The formula for the binomial distribution is given by,

p( X=x )= C n a p n−x q x

Probability of at least 4 successes P( x≥4 ) is given by,

P( x≥4 )=P( x=4 )+P( x=5 )+P( x=6 ) = C 6 4 p 6−4 q 4 + C 6 5 p 6−5 q 5 + C 6 6 p 6−6 q 6

Substitute the values of p and q in the above expression.

P( x≥4 )= C 6 4 ( 1 3 ) 6−4 ( 2 3 ) 4 + C 6 5 ( 1 3 ) 6−5 ( 2 3 ) 5 + C 6 6 ( 1 3 ) 6−6 ( 2 3 ) 6 =15× ( 1 3 ) 2 ( 2 3 ) 4 +6× ( 1 3 ) 1 ( 2 3 ) 5 +1× ( 1 3 ) 0 ( 2 3 ) 6 =15× ( 2 ) 4 ( 3 ) 6 +6× ( 2 ) 5 ( 3 ) 6 +1× ( 2 ) 6 ( 3 ) 6 = ( 2 ) 4 ( 3 ) 6 [ 15+12+4 ]

Further simplify the above expression.

P( x≥4 )= ( 2 ) 4 ×31 ( 3 ) 6 = 31 9 ( 2 3 ) 4

Thus, the probability of at least 4 successes is 31 9 ( 2 3 ) 4 .