Question

The probability of success is twice of probability of failure . Find the probability of at least 4 sucesses in next 6 trials

Answer 1

Given that probability of success is twice the probability of failure.

Let probability of failure p = x

So probability of success p= 2x

probability of failure + probability of success = 1

=> x + 2x = 1

=> 3x = 1

=> x = 1/3

So p = 1/3 and q = 2/3

Let X is arandom variable thet represents the number of success in six trials.

So P(X = x) = nCx * pn-x * qx 

Now probability of atleast 4 success = P(X>=4)

                                                       = P(X = 4) + P(X = 5) + P(X = 6)

                                                       = 6C4 * (1/3)6-4 * (2/3)4 + 6C5 * (1/3)6-5 * (2/3)5 + 6C6* (1/3)6-6 * (2/3)6  

                                                       = 15* (1/3)2 * (2/3)4 + 6 * (1/3) * (2/3)5 +  * (2/3)6                                  

                                                       = (2/3)4 * {15/9 + 12/9 + 4/9}

                                                       = (2/3)4 * {31/9}

                                                       = (16*31)/(81*9)

                                                       = 496/ 729