A Bernoulli process has 4 trials and probability of success 0.31. Find the following probabilities. 1. Exactly 2 successes. 2. Exactly 2 failures. 3. At most 1 success. 4. At least 1 success. 5. At least 1 success and 1 failure.

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A Bernoulli process has 4 trials and probability of success 0.31. Find the following probabilities.  1. Exactly 2 successes.  2. Exactly 2 failures.  3. At most 1 success.  4. At least 1 success.  5. At least 1 success and 1 failure.

Nicole Conner · Accepted Answer

Step 1  Given Information:  A Bernoulli process has 4 trials and probability of success is 0.31.   A sequence of Bernoulli trials is the binomial probability.  The binomial probability is the probability of exactly x successes on n repeated trials and X can only have two outcomes.  Formula:P(X=x)=nCxpx(1−p)n−x  We have n=4 and p=0.31  (1) Exactly 2 successes:  Required probability is obtained as follows:  P(X=2)=4C2(0.31)2(1−0.31)4−2 NKS =4!2!(4−2)!×0.0961×0.4761  =4×3×2!2×1×2!×0.04575321  =6×0.04575321  =0.27451926  ≈0.2745  Step 2  (2) Exactly 2 failures:  This is 2 failures and 4−2=2 successes.  Required probability is obtained as follows:  P(X=2)=4C2(0.31)2(1−0.31)4−2  =4!2!(4−2)!×0.0961×0.4761  =4×3×2!2×1×2!×0.04575321  =6×0.04575321  =0.27451926  ≈0.2745  (3) At most 1 success:   Required probability is obtained as follows:  P(X≤1)=P(X=0)+P(X=1)  =4C0(0.31)0(1−0.31)4−0+4C1(0.31)1(1−0.31)4−1  =4!0!(4−0)!×1×0.22667121+4!1!(4−1)!×0.31×0.328509  =1×1×0.22667121+4×0.10183779  =0.63402237  ≈0.6340  Note:   Since, there are multiple subparts, we have solved first three parts for you. To get remaining subparts solved, please repost the complete question and mention the parts to be solved.

A Bernoulli process has 4 trials and probability of success

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