Bernoulli Trials and Binomial Distribution - Practice Questions & MCQ

Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions 

1. There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.

2. The n trials are independent and are repeated using identical conditions. 

3. There are only two possible outcomes, called "success" and "failure," for each trial. The letter p denotes the probability of a success on any one trial, and q denotes the probability of a failure on any one trial. p + q = 1

For example, randomly guessing at a true-false statistics question has only two outcomes. If a success is guessing correctly, then a failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true-false question with a probability p = 0.6. Then, q = 0.4. This means that for every true-false statistics question Joe answers, his probability of success (p = 0.6) and his probability of failure (q = 0.4) remain the same. So guessing one question is considered a trial. If he guesses n different questions, means the trial is repeated n times and p = 0.6 remains the same for each trial.

Binomial Distribution

Let an experiment has n independent trials and each of the trial has two possible outcomes i.e. success or failure. If getting a number of successes in the experiment is a random variable then probability of getting exactly r-successes is -

Where P(X=r) is the probability of X successes in n trials when the probability of success in ANY ONE TRIAL is p. And of course q=(1-p) and is the probability of a failure in any one trial.

In the experiment, the probability of 

• At least “r” successes, 

• At most “r” successes, 

A binomial distribution with n-Bernoulli trials and probability of success in each trial as p, is denoted by B(n, p).

The mean, μ, and variance, σ² , for the binomial probability distribution are

μ = np     and     σ² = npq

The standard deviation, σ, is then

.