Join Eddie Davila for an in-depth discussion in this video Binomials, part of Statistics Foundations: Probability.
- When an experiment has only two possible outcomes, we call this a binomial random variable. A coin flip can only result in heads and tails. Eligible voters can either vote or not vote. A patient can either test positive or negative for a disease. These are possible binomial random variables provided we have n trials with a probability of success we call p. Typically in stats, n is the number of instances. In other words, if we take our coin and flip it four times, n is equal to four, the number of flips, p is equal to 0.5, the chance of success which can be either heads or tails in this case. How do we take voter turnout? Suppose there are 5,000 registered voters, and let's say the probability a registered voter will actually vote is 60%. In this situation, n is equal to 5,000 and p is equal to 0.60 or 60%. Let's use binomials to solve a problem. Suppose an organization has a monthly meeting. New people attend the meeting each month, but only 20% end up joining the organization. Suppose three people attend this month's meeting: Harry, Amani and Luna. What are the chances one of these three people will join the organization? It's important to clarify your question. What are the chances that one, not more than one, not less than one, exactly one will join the organization? Well, we have three ways this can happen. Here's the first way. Harry joins, Amani does not join, Luna does not join. There's a 20% chance Harry joins, and since there is a 20% chance Amani will join, that means there's an 80% chance that she will not join. The same goes for Luna. If we multiply 0.2 times 0.8 times 0.8, we get 0.128 or 12.8%. Remember though, we can also achieve success if Amani joins, but Luna and Harry do not join. That also results in one person joining the organization. Finally, Luna could join and Amani and Harry would not join. Each scenario has a 12.8% chance of occurring. And if we add those probabilities, we end up with our answer. There is a probability of 38.4% that only one of our three new people at this month's meeting will join. Using this same method, we could figure out the probability that exactly zero, one, two, or all three of our new meeting attendees would join our organization. Here's the chart with those answers including the one we just calculated for one person out of three joining. The calculations for tougher binomial problems can get really ugly really fast. As a result, many folks use binomial probability tables. Here's just a tiny portion of one and it happens to include the answer we just calculated. Along the left, see n, the number of trials. We had three. Then across the top, we find our p, the probability. Remember, the probability of success for each person was 20% or 0.20. So, where n equals three and p equals 0.20, we find our four numbers provided: 0.512, 0.384, 0.096, and 0.008. As you can see, the chart gets very ugly and it only covers a very limited range of ns. So, what happens when n gets big? Suppose n is equal to a million. These charts wouldn't be in much use, but luckily when n gets bigger, we can use calculus. Probably not what you wanted to hear. So, here's some better news. In our binomial experiments, when p is equal to 0.50 and when n gets very big, our distribution result is a normal distribution. What about when p is not equal to 0.50? Good news here too. In most cases, when n our number of trials gets really big, the asymmetry of our non-equal probability is overwhelmed and the resulting distribution can be approximated with the normal curve.
Eddie explains that probability is used to make decisions about future outcomes and to understand past outcomes. He covers permutations, combinations, and percentiles, and goes into how to describe and calculate them. Eddie introduces multiple event probabilities and discusses when to add and subtract probabilities. He describes probability trees, Bayes’ Theorem, binomials, and so much more. You can learn to understand your data, prove theories, and save valuable resources—all by understanding the numbers.