## Statistics – Distributions

**Statistics – Distributions**

**Binomial Distribution** is a discrete **probability distribution** that applies when an experiment is conducted n times with each trial having a probability p of success and each trial being independent of every other trial. One can think of an experiment that has only two possible results: success or failure. The probability of success for each trial is p, and the probability of failure is 1-p. if the experiment is conducted 10 times, how many trials should result in success? If the probability of success p equals 0.8, then it is obvious to expect that there would be 8 successes.

To make the desired approximation, we need to take into account one major difference between the binomial and the **normal probability distribution**. The binomial random variable is discrete, whereas the normal random variable is continuous. A failure or a success of an experiment is called Bernoulli’s experiment or Bernoulli’s trial, where n =1, the **Binomial Distribution** is a **Bernoulli Distribution**.

If the experiment is conducted just 2 times, what is the probability that both the trials will result in success. If A is the event of getting a success on the first trial and B is the event of getting a success on the second trial, then Pr (A) = p and Pr (B) = p. The event of getting successes on both trials can be written as A ∩ B. This means that the chance of getting a success on any particular trial is not affected by the results of any of the other trials. If the trials are independent, then one can multiply the two probabilities:

Pr (A ∩ B) = Pr (A and B) = Pr (A) Pr(B) = p^{2}

Binomial formula: Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success of an individual trial is P, then the binomial probability is

b (x;n,p) = _{n}c_{x}× p^{x} (1-p)^{n-x}

Suppose that if a die is tossed 5 times. What is the probability of gettingexactly 2 fours

This is a binomial distribution experiment, where the number of trials are 5, the number of successes are 2, and the probability of success on a single trial is 1/6 or about 0.167.

b (2; 5, 0.167) = _{5}C_{2} × (0.167)^{2 }× (0.833)^{3}

b (2; 5, 0.167) = 0.161

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Our previous articles on Statistics include** Probability in Statistics, Using SPSS for Statistics, Hypothesis Testing, Correlation in Statistics**