 So, let's talk about what's called the normal distribution. Although there are several types of probability distributions, the most important is the normal distribution. And this is defined using a rather frightening formula. But the important ideas are the following. First of all, the graph of this normal probability distribution looks like this, and it's sometimes referred to as the bell curve. Now, let's take a look at some of the important features of the normal distribution. The center corresponding to the highest point occurs at the mean. This symbol comes from calculus, and what it tells us is that this probability that we're between A and B is going to be determined by finding the area under the curve between A and B. So this means that when finding probabilities of normally distributed outcomes, it is very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very helpful to draw a picture of the area corresponding to the probability. While computing the exact value of a normally distributed probability is best left to a calculating device, there are some useful benchmarks. The probability of being within one standard deviation of the mean is 68%. So if we take our bell curve, mark the mean, and mark one standard deviation above, and one standard deviation below, the area between these two lines is 68% of the total area. And, since the area is the probability, that says the probability we're within one standard deviation is 68%. Likewise, the probability of being within two standard deviations of the mean is 95%. And so again, here's our mean. Two standard deviations below is some place around here, and two standard deviations above is some place around here. The probability is the area, so the area of the central region is 95% of the area under the curve. And finally, the probability that we're within three standard deviations of the mean from here, all the way up to here, is 99.7%. This observation allows us to compute some normal probabilities right away. For example, suppose the weight of cereal in a box is normally distributed with a mean of 16 ounces, and a standard deviation of 0.2 ounces. Of 1,000 boxes, how many can be expected to contain between 15.8 and 16.2 ounces? So to begin with, we might graph our normal distribution with a mean of 16 ounces, and we can graph this as follows. All normal distribution curves look essentially the same. The only difference between the normal distribution curves is where the mean is located, and in this case, the mean is at 16, and that's our high point of the curve. Now, we want to be between 15.8 and 16.2 ounces, so let's mark those locations. 15.8 is a little bit below the mean, and 16.2 is a little bit above the mean. Now, if our cereal box contains between 15.8 and 16.2 ounces, it falls someplace within this interval, which means we want to know the area of this region. Now, since our standard deviation is 0.2 ounces, that means that 15.8 and 16.2 represent all values within one standard deviation of the mean. By our 68.9599.7 percent observation, we know that this area is about 68 percent of the total area, and so the probability we followed this interval is 68 percent. Now, by the frequentist interpretation of probability, this means we can expect about 68 percent of our boxes to contain this much cereal, and 68 percent of 1,000 is 680. Another important observation about the normal distribution is that it's symmetric about the mean, and what this tells us is the following. If A is k units above the mean, and B is k units below the mean, then the probability that a random variable is equal to A is the same as the probability a random variable is equal to B. So, same setup, the weight of cereal is normally distributed with the mean of 16, and a standard deviation of 0.2 of 1,000 boxes, how many can be expected to contain more than 16 ounces. Since we know the mean, and the normal distribution is symmetric about the mean, we can sketch a picture of the normal distribution. The high point is located at the mean 16 ounces, and since the probability corresponds to the area, we can identify the area we need to find. Since we're interested in a cereal that contains more than 16 ounces, we want this area to the right of the mean. Since the normal distribution is symmetric about the mean, that means half the area will be above the mean, and half the area will be below the mean. So the part above the mean has area 1 divided by 2, 0.5, or 50%. And so the probability of the weight in the cereal box is more than 16 ounces, is 50%, and so we can expect 50% of 1,500 boxes of cereal to have weights in this interval. So once again, let's suppose that the weight of cereal in the box is normally distributed with the mean of 16 ounces, and a standard deviation of 0.2 ounces of 1,000 boxes, how many can be expected to contain between 15.8 and 16 ounces? So since we know the mean, and the normal distribution is symmetric about the mean, we can sketch a picture of the normal distribution. No, no, draw the normal distribution. Since the probability corresponds to the area, we can identify the area we need to find. We want the weight to be between 15.8 and 16 ounces, and so that means we want the area of this region starting at 15.8 and ending at 16. So notice that 15.8 is one standard deviation below the mean, and we know that the probability of being within one standard deviation of the mean is 68%. Which means the area from 15.8, that's one standard deviation below, to 16.2, that's one standard deviation above, will be 68%. Since the normal distribution is symmetric about the mean, this area is split in half by the mean. So the part above the mean has area 0.68 divided by 2, 0.34, or 34%, and the part below the mean has the same area. And so the probability the weight of the cereal in the box is between 15.8 and 16 ounces is 34%. And so we can expect that 34% of 1,340 of the boxes of cereal to have weights in the interval 15.8 to 16 ounces.