The most important probability distribution for discrete random variables is the binomial distribution. It gives probabilities for accounts with binary data. Because there are so many situations with binary data, it's used frequently. In this video, I'll explain the most important properties of the binomial distribution. And also how to apply it in probability calculations. Let's start off with a few examples where you have a kind of chance experiment with two outcomes. Participants in a meeting are on time or too late. Voters are in favor or against a proposal, or the noise level during a concert exceeds 80 decibels or not. When you collect a given number of observations and such phenomena or trials, the number of cases where you get one of the two outcomes, the number of success often follows a binomial distribution. For example, you could consider how many, out of 25 participants in a meeting, are late, or how many of ten voters vote against a proposal. There are two conditions to be met before you can be certain that a random variable follows a binomial distribution. The first is that the probability of success in each separate trial is the same throughout the experiment and a second is that trials are statistically independent. This means that the result of one trial does not depend on the results of others. In fact, you have now already encountered the three ingredients for a binomial distribution. First, there is a phenomenon or trial with two possible outcomes and a constant probability of success. This is called a Bernoulli trial. Second, you observe the outcome of the trial n times. And third, you count the number of successes, x. These three elements are combined in one formula that gives the probability of getting a particular number of successes with n trials. The formula is shown here. You can just fill in the three numbers, n, x, and p to get the answer. As indicated, on the second line below the formula, the binomial distribution is a discrete distribution where the random variable x can only take the values ranging from zero up to n. Which makes sense, as you can only have a finite number of successes. Zero, one, two, up to the number of n trials. Therefore the formula is a probability mass function. It gives the probability matching with each possible value of x and you don't need to consider an interval as an inner probability density function. This symbol, the exclamation mark is not very frequently used, it's called factorial and it's shorthand for multiplication or for all integers up to the number specified. For example, four factorial is shorthand for one times two times three times four. This entire first term in the formula gives the number of ways you can select x elements, disregarding their order, from a set of n elements. It's also called the binomial coefficient and is sometimes written in this way. The short end for the entire formula is this, which says x is a binomial random variable with n trials and success probability p. Now let's apply the binomial formula to a specific example. Once everyday you travel along a route where you have to pass a bridge. The bridge is open 10% of the time but the exact moment of its opening are random. What standard probability that you would encounter an open bridge on 0, 1, 2, up to 5 days during a given week? Your experiment has five trials, and you have a probability of 0.1 to encounter an open bridge. So, the binomial distribution, in this case, is this, with x the number of times you encounter an open bridge. By filling in values 0, 1, 2, up to 5 for x, you get the following probabilities. And if you sum the six probability values, you find that equals one, which had to be the case, because the outcomes from x equals zero to five form the set with all possible outcomes for this random variable. Okay, let's move on to a related question, using the same example. What would be the probability to encounter an open bridge on at most one day over a period of five days? Here, we can make good use of the probability table that was just created. We are looking for the total probability of the case where the bridge was never opened, or opened on just one day, so the sum of these two probabilities, which is 0.92. To answer this last question, we make use of the cumulative binomial probability distribution. Given the total probability of all outcomes lower than or equal to a given number of successes. The equation of this cumulative probability distribution is as follows. It's almost identical to the binomial probability mass formation, but now with the summation in front and all x's replaced with the symbol k, to increment the value for the number of successes from zero up to x in the summation. Let's now look at the shape of a binomial distribution. It's discrete, meaning that it only gives probabilities for axis zero, one, two, etc. And also that it's bounded between 0 and n, the number of trials that you are considering. Interestingly, the shape of the binomial distribution can change considerably for variations of the parameter, p, the probability of success. Depending on this parameter, the distribution can vary between right-skewed to symmetric and to left-skewed. These three distributions show the numbers of successes in 20 trials for different probabilities of success. For the first, the probability of success is 0.1, for the second it's 0.5, and for the third it's 0.9. In general, a binomial distribution with a low probability of success is right skewed. While that with a high probability of success is left skewed. By aligning the distributions horizontally, you can see that the peak of the middle one is lower that the others, so it's more spread out. This an interesting property of the binomial distribution. Its standard deviation depends on the value of p, and so does its mean, as a matter of fact. The mean of a binomial distribution is equal to n times p. And its standard deviation is equal to the square root of n times p times one minus p. For p is zero or one, the standard deviation is zero. And for p is a half, it reaches a maximum or 0.5 times the square root of n. Let me summarize what I have explained in this video. The binomial distribution is a discrete probability distribution that is used when a random variable can have two mutually exclusive outcomes, success and failure. It gives the probability of observing x successes in n outcomes of the random variable, so called trials, with the probability of success on a single trial denoted by p. The binomial distribution assumes that p is fixed for all trials. The mean of such a distribution is n times p and its standard deviation is the square root of n times p times 1 minus p. The shape of the binomial distribution varies from right-skewed when the value of p is close to zero to symmetric when p is around 0.5 and left-skewed when p is close to one. Here the formula for the binomial probability distribution is shown. And this is the same formula in short hand. Finally, the formula for the cumulative probability distribution is shown here.