Binomial experiments: When there are only two possible outcomes

- When an experiment has only two possible outcomes, the result is what we call 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. 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 either can be heads or tails in this case.

How 'bout if we take voter turnout? Let's say there are 5,000 registered voters. And let's say the probability a registered voter will actually vote is 60%. Now, n equals 5,000 and p is equal to 0.6, 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, Tom, Lori, and Fred. What are the chances one of those three people will join the organization? It's important to clarify our question. What are the chances that one, not more than one, not less than one, will join the organization? Well, we have three ways this can happen. Here's the first way this can happen. Tom joins. Lori does not join. Fred does not join. There is a 20% chance Tom joins.

And since there is a 20% chance Lori will join, that means there is an 80% chance she will not join. The same goes for Fred. 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 Lori joins but Fred and Tom do not join. That also results in one person joining the organization.

Finally, Fred could join and Lori and Tom 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 38.4% chance that only one of our three new people at this month's meeting will join. Using the 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 a 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, we see n, the number of trials. We had three. Then, across the top we find our p.

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: 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 n's.

So what happens when n gets big? Suppose n equals a million. These charts wouldn't be 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 0.5, and when n gets very big, our distribution results in 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.