Question

Question: Find each of the following probabilities when $\mathit{n}$independent Bernoulli trials are carried out with probability of success$\mathit{p}$.

a) The probability of no failures

b) The probability of at least one failure

c) The probability of at most one failure

d) The probability of at least two failures

Short answer

Answer:

a) The probability of no failure is$p^n$

b) The probability of at least one failure is $1-p^n$.

c) The probability of at most one failure is$p^n+n{\left(p\right)}^{n-1}\left(1-p\right)$.

d) The probability of at least two failures is $1-\left[p^n+n{p}^{n-1}\left(1-p\right)\right]$.

Step-by-step solution

Given Information

It is given that $n$independent Bernoulli trials are carried out with probability of success$p$

Definition and formula to be used

A Bernoulli trial is an experiment which consists of two outcomes, the one with probability $\mathbf{}\mathit{p}$is known as Success and the other one with probability $\mathit{q}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{}\mathbf{-}\mathbf{}\mathit{p}$is known as Failure.

Find the probability of no failure

Given that $n$independent Bernoulli trial are carried out with probability of success$p$

The probability of failure is $1-p$.

The probability of no failure is same as the probability of all success,

${}^n{C}_n{\left(p\right)}^n{\left(1-p\right)}^0=p^n$

The probability of no failure is $p^n$.

Find the probability of at least one failure

Given that $n$independent Bernoulli trial are carried out with probability of success$p$

The probability of failure is $1-p$.

The probability of at least one failure is equal to subtraction of no failure from total probability ,

$1-{}^n{C}_n{\left(p\right)}^n{\left(1-p\right)}^0=1-p^n$

The probability of at least one failure is $1-p^n$.

Find the probability of at most one failure

Given that $n$independent Bernoulli trial are carried out with probability of success$p$

The probability of failure is$1-p$.

The probability of at most one failure is equal to the sum of no failure and one failure probabilities,

${}^n{C}_n{\left(p\right)}^n{\left(1-p\right)}^0+{}^n{C}_n{\left(p\right)}^{n-1}{\left(1-p\right)}^1=p^n+n{\left(p\right)}^{n-1}\left(1-p\right)$

The probability of at most one failure is $p^n+n{\left(p\right)}^{n-1}\left(1-p\right)$.

Find the probability of at least two failures

Given that $n$independent Bernoulli trial are carried out with probability of success$p$

The probability of failure is$1-p$.

The probability of at least two failures is equal to subtraction of one and zero failure from the total probability ,

$1-\left[{}^n{C}_n{\left(p\right)}^n{\left(1-p\right)}^0+{}^n{C}_{n-1}{\left(p\right)}^{n-1}{\left(1-p\right)}^1\right]=1-\left[p^n+n{p}^{n-1}\left(1-p\right)\right]$

The probability of at least two failures is$1-\left[p^n+n{p}^{n-1}\left(1-p\right)\right]$.

Thus,

a) The probability of no failure is$p^n$

b) The probability of at least one failure is $1-p^n$.

c) The probability of at most one failure is$p^n+n{\left(p\right)}^{n-1}\left(1-p\right)$.

d) The probability of at least two failures is$1-\left[p^n+n{p}^{n-1}\left(1-p\right)\right]$.