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Frauffshiesiaf6s

Answered question

2022-02-13

How do you find the probability of at least one success when n independent Bernoulli trials are carried out with probability of success p?

Answer & Explanation

sigcinweq0s

Beginner2022-02-14Added 19 answers

The probability is

P (X = k) = (\begin{matrix} n \\ k \end{matrix}) p^{k} q^{n - k}

where,

q = 1 - p

Therefore,

P (X = k) = (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k}

In our case,

k \geq 1

So,

P (X = 0) = (\begin{matrix} n \\ 0 \end{matrix}) p^{0} q^{n - 0}

(\begin{matrix} n \\ 1 \end{matrix}) = \frac{n!}{1! (n - 1)!} = n

P (X = 0) = {(1 - p)}^{n}

P (X \geq 1) = 1 - P (X = 0) = 1 - {(1 - p)}^{n}

Ethen Welch

Beginner2022-02-15Added 11 answers

To calculate the probability of getting at least one success you use opposite event formula.
The event opposite to given is You got no success in n trials. Therfore the probability is:

P (k = 0) = (\begin{matrix} n \\ 0 \end{matrix}) (p^{0}) (1 - p)^{n} = (1 - p)^{n}

So the probability we are looking for is:

P (k \geq 1) = 1 - P (k = 0) = 1 - {(1 - p)}^{n}

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