 There are a number of important probability distributions, but one of the more common is known as the binomial distribution. In many cases, we can describe a random experiment as the repetitions of a single, simpler experiment. So, for example, flipicoin ten times is the same as flipicoin once, repeat ten times. Or, roll three dice is the same as roll one die, three times. If this happens, we have a binomial experiment. More formally, we say that the binomial experiment is one that can be described as a single, simple experiment, repeated a specified number of times, where the probability of an event of interest is constant. So, let's determine if the following experiment is binomial, a coin is flip ten times, and the number of heads is recorded. If it's binomial, we do want to identify this simple experiment and the probability, otherwise we want to explain why it's not binomial. So, we'll pull in our definition of binomial experiment. So, we need to be able to describe this as a single, simple experiment, repeated a specified number of times. So, this is the experiment, flip a coin one time, repeated ten times. The other requirement for being a binomial experiment is that the probability of an event of interest is constant. Here, we're interested in recording the number of heads, so that means the event of interest is that the coin lands heads, and the probability a coin lands heads on any individual flip is one-half. And since this probability doesn't change, this is a binomial experiment. How about this experiment? We roll three dice, and the sum of the numbers is found. Now, we can treat this as the experiment, roll one die, repeated three times. But, there's no event, roll one die, that corresponds to the sum of the numbers. And that means we can't even identify the event of interest, let alone determine whether the probability is constant. And so, this is not a binomial experiment. Or, suppose we pick three people from a group of eight men and two women to serve on a committee, and the number of women is recorded. If this is a binomial experiment, identify the simple experiment of the probability, otherwise explain why it's not binomial. So let's go to the worst possible answer. This is not a random experiment, because the people are chosen. And, while that's true, the thing to remember is that random means unpredictable, not reasonless. Since we don't know in advance who is going to be chosen, this is a random experiment. So let's see if we can describe it as a simple experiment repeated a number of times. And the first thing that comes to mind is we might try to describe this as the experiment, pick one person from a group of ten repeated three times. But, we'd be wrong. After we pick the first person, we no longer have ten people to choose from. And so, this is not a binomial experiment. So how do we calculate binomial probabilities? If a binomial experiment is repeated n times where an event of interest has probability p, the probability the event will occur k times is given by the following formula. Now, as a general rule, calculating the different components of the formula is somewhat tedious, and as a result, most spreadsheet, statistical programs, and calculators have a binomial distribution functions. Syntax varies, but binomedist n k p is common. And note that what this means is that once we identify that an experiment is binomial, what we have to know is how many times the experiment was repeated, how many times the event of interest occurred, and the probability of the event of interest. So let's find the probability that a fair coin lands heads eight times in ten flips. So note that we're interested in the number of occurrences of an event when a simple experiment is repeated a number of times where an event of interest has constant probability. And so this means we're dealing with a binomial experiment and can compute a binomial probability. In order to do that, we need to identify the number of times the experiment is repeated, the probability of the event of interest, and the number of occurrences of the event of interest. So we're interested in the probability that a fair coin lands heads eight times in ten flips. And so that says the number of occurrences of the event is k equal to eight. Our simple experiment is flip a coin once, and we've repeated that experiment n equals ten times. Now, we also need the probability of the event of interest. Well, we're interested in whether the coin lands heads, and so the probability of landing heads is one-half. So this is a binomial experiment with p equals one-half, k equals eight, n equals ten, and so we can compute the probability. Again, most spreadsheet calculators and statistical software have a built-in capability of calculating binomial probabilities, you just have to figure out how. Once you identify n, k, and p, and we have, we'll enter in the correct information and our spreadsheet will calculate the probability for us.