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) = p2
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) = ncx× px (1-p)n-x
Suppose that if a die is tossed 5 times. What is the probability of getting exactly 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) = 5C2 × (0.167)2 × (0.833)3
b (2; 5, 0.167) = 0.161
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