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Bernoulli Trials and Binomial Distribution is considered one the most difficult concept.
54 Questions around this concept.
A multiple-choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct.The probability that a student will get 4 or more correct answers just by guessing is:
Consider 5 independent Bernoulli's trials each with probability of success . If the probability of at least one failure is greater than or equal to
, then
lies in the interval
An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is:
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Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions
There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
The n trials are independent and are repeated using identical conditions.
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
.
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