How to Find Maximum Likelihood Estimation

Imagine that in a package of light bulbs, seven out of 10 bulbs work and that this package was part of a large shipment of other similar packages. The "common sense" way to figure out the proportion of working light bulbs from this sample is to divide 7/10 and estimate a population proportion of .7 working bulbs.

Rather than leave matters at that, think of all the possible sample proportions. MLE takes this approach because there is only a likelihood that the entire population has a proportion of .7 working bulbs as the example sample did.

Use a binomial calculator or Individual Binomial Probabilities table to derive the likelihood of all possible proportions. Enter "10" in the "n" field and "7" in the "Prob. X" field.

Test the likelihood of different probabilities of success by placing different values in the "p" field. If you enter ".2" in the "p" field on the binomial calculator, for example, you will see that the likelihood of getting the observed sample of 7/10 working bulbs is only .0008. What this means, in other words, is that if each bulb has only a .2 probability of working, the chance of getting seven working bulbs in a sample of 10 is very low.

Try to see which proportion of success value yields the highest probability, given an n of 10 and a Prob. X of 7. You should see that a p of .7 produces this result, with a probability of .267. This finding indicates that you would receive the a package of bulbs exactly like the example sample a little over one quarter of the time. As a general rule, the MLE of the binomial proportion should equal the common sense proportion, in our case .7.