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Binomial DistributionPerhaps the most widely known of all discrete distributions is the binomial distribution. The assumptions underlie the distribution is (I) The experiment involves n identical trials.(ii) Each trial has only two possible outcomes denoted as success or as failure.(iii) Each trial is independent of the previous trials.(iv) The terms p and q remain constant throughout the experiment, where the term p is the probability of getting a success on any one trial and the term q=1-p is the probability of getting a failure on any one trial.The probability of r successes in n trials with probability p for a success in each trial is given by P(r, n, p)= where p+ q=1rnrqprnrn)!(!!
Some example of the Binomial Distribution:1.Suppose a machine producing computer chips has a 6% defective rate. If a company purchases 30 of these chips, what is the probability that none is defective?2.Suppose that the brand X car battery has a 35% market share. If 70 cars are selected at random, what is the probability that at least 30 cars have a brand X battery.
Examples: At Kerr pharmacy, where employees are often late. Five workers are in the pharmacy. The owner has studied the situation over a period of time and has determined that there is a 0.4 chance of any one employee being late and that they arrive independently of one another. How would we draw a binomial probability distribution illustrating the probabilities of 0,1,2,3,4,or 5 workers being late simultaneously?
Observation: 1. When p is small (0.1), the binomial distribution is skewed to the right.2. As p increases (to 0.3, for example), the skew ness is less noticeable.3. When p=0.5 the binomial distribution is symmetrical.4. When p is larger than 0.5, the distribution is skewed to the left.5.The probabilities for 0.3, for example, are the same as those for 0.7 except that the values of p and q are reversed. This is true for any pair of complementary p and q values (0.3 and 0.7, 0.4 and 0.6, and 0.2 and 0.8)