Thursday, May 9, 2013

Some Properties of the Binomial Distribution

The main focus of a binomial experiment is to find the probability of r successes in n trials. How are these probabilities calculated? How is mean and standard deviation of a binomial distribution calculated? How is a binomial distribution displayed graphically?

Suppose you wish to flip a fair coin three times. What is the probability of obtaining two heads in the three tosses? This is an example of a binomial experiment, with probability of success 0.5. We could list the possibilities as {HHH, HHT, HTH, THH, HTT, TTH, THT, TTT}. You notice from the sample space that the probability is 3 out of 8, or 3/8. But most times it's much too time consuming to list all possibilities. So there is a formula that we can use:

P(r) = n!/[r!(n - r)!]* prq(n-r)

where n = number of trials
p = probability of success on a trial
q = probability of failure on a trial
r = random variable representing the number of successes out of n trials

Using the coin tossing example above, P(0) = 1/8, P(1) = 3/8, P(2) = 3/8, P(3) = 1/8.

To graph a binomial distribution, place the r values on the horizontal axis. Next, place the P(r) values on the vertical axis. Then construct a bar over each r value. Extend these bars to r - 0.5 to r + 0.5. The height of each bar must be P(r).

In the above example the bar widths would be -0.5 to 0.5, 0.5 to 1.5, 1.5 to 2.5, and 2.5 to 3.5. You cannot have a 0.5 of a success but to have a bar, there needs to be a width, so you add and subtract 0.5 from each value of r.

To help describe the graph of the binomial distribution, the mean and standard deviation are very helpful. For this distribution, the mean is np and the standard deviation is the square root of npq.

In our example with the coins, the mean is 3(0.5) = 1.5, and the standard deviation is square root of 3(0.5)(0.5) = 0.87.

This simple guide should help students understand some properties of the binomial distribution including calculating probabilities, graphing, and computer the mean and standard deviation.

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