 Remember that given an event in a null hypothesis, the p-value corresponds to the probability of observing an event of equal or lesser probability under the assumption the null hypothesis is true. This means we have to be able to compute probabilities and select those corresponding to events of equal or lesser probability than the observed event. For example, suppose a coin is flipped ten times and observed to land heads eight times. If we reject the null hypothesis, the coin is fair on the basis of this observation, what is the corresponding p-value, and interpret the result. So note that we have the number of occurrences of an event. We know that the coin landed heads k equals eight times. When a single experiment, flip a coin, is repeated a number of times, n equals ten, and an event of interest has a constant probability. The probability of the coin lands heads is always one-half, so our observation comes from a binomial experiment. And this means we can calculate that binomial probability using a formula or technological device. Since we're trying to calculate a lot of probabilities, we might want to use our technology, but remember technology is a power tool, so be very careful with how you use it. Since the coin can land heads anywhere between zero and ten times in ten flips, we compute these probabilities. So after we've calculated the probabilities, we note first of all, if the null hypothesis is true, the probability of the observed outcome, eight heads in ten flips, is 0.0439. And now we'll look at the table of probabilities for any outcome that has an equal or lower probability. And those are going to be these outcomes. The outcomes of equal or lower probability are getting zero, one, two, eight, nine, or ten heads. And the p-value is the sum of the probabilities of these events, so we'll take all of these probabilities and add them up, and so now let's interpret our result. This probability, 0.1094, this is the probability a fair coin would have produced an observation as or less likely. And so we might say that it's unlikely a fair coin would have produced the observation landing heads eight times in ten flips. It's very important to remember the p-value is not the probability that we have made an error in our decision. What if we want to look at a problem that may have a little bit more importance than whether a coin is fair or not, so knows a firm has twelve employees, seven of whom are women. Two employees are chosen to serve on a committee and both are women. On the basis of this observation, if we reject the null hypothesis gender played no role in the selection, what is the p-value, and let's interpret the significance. So let's take a few notes. We've selected a specific number, two employees, from a fixed set, twelve employees, where a fixed number have a specific property, seven are women. And so this is a hypergeometric experiment, and we can calculate the probabilities using a formula or our technological device. And again, since we do have to compute several probabilities, our technological device may be a better solution. So let's take a closer look. If the null hypothesis is true, the probability of the observed outcome to women is 0.3182. Looking at our table, we see that there's one event of equal or lower probability, zero women, probability 0.1515, and the p-value is the sum of the probabilities of the events of equal or lower probability. So our p-value is going to be 0.4697. And this is the probability that an bias selection would yield the observation to women on the committee. It's reasonably likely that an unbiased selection would yield the observation. Or we might have another example. A manufacturer claims that a spool of wire has fewer than two defects per 100 meters, and we test a 100-meter section, and we find three defects. Let's find the corresponding p-value and its significance. The event is rare. There are two defects per 100 meters, but it could potentially happen an infinite number of times. Why don't we use a Poisson distribution? The mathematical one. Now, because our event could occur an infinite number of times, we'll have to make some changes in how we approach the problem. But we start out the same way by calculating a bunch of probabilities. And since we expect lambda equals 2 defects in 100 meters, we can compute the probabilities. Now since the event could occur an infinite number of times, we have to leave the last probabilities uncomputed. So if there are an average of two defects per 100 meters, the probability of the observed event, 3 defects, is 0.1804. And the events of equal or lower probability are having zero defects, or having three or more defects. Because remember, we could have an infinite number of defects. Now the p-value will be the sum of these probabilities. And so we know the probability of having zero defects. Now to find this other probability, remember the event 3 or more defects is complementary to the event 0, 1, or 2 defects. Well we know those probabilities. The probability of 0, 1, or 2 defects will be, and so the probability of 3 or more defects is going to be, and so we can calculate our p-value, and that's about 46 percent. So it's likely that a wire with an average of 2 defects per 100 meters could produce the observed result.