Question

Question:

(a) What is meant by a Bernoulli trial?

(b) What is the probability of\(k\)successes in\(n\)independent Bernoulli trials?

(c) What is the expected value of the number of successes in\(n\)independent Bernoulli trials?

Short answer

Answer

(a) A Bernoulli trial is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted.

(b) The resultant answer is \(\left( {\begin{aligned}{{}{}}n\\k\end{aligned}} \right){p^k}{(1 - p)^{n - k}}\).

(c) The expected number of successes in \(n\) independent Bernoulli trials is \(np\) where \(p\) is probability of success.

Step-by-step solution

Given data

The given data is \(k{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n\) elements.

Concept of Probability

Finding the probability of an event occurring can be thought of as probability.

Concept of Bernoulli trial

(a)

A Bernoulli trial is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted.

Find the probability of a Bernoulli trial

(b)

The probability of success of a Bernoulli trial is given by \(\left( {\begin{aligned}{{}{}}n\\k\end{aligned}} \right){p^k}{(1 - p)^{n - k}}\) .

Simplify the expression

(c)

The expected number of successes in \(n\) independent Bernoulli trials is \(np\) where \(p\) is probability of success.