Probability Distribution

A probability distribution is a graph or table that pairs all the possible outcomes of some random variable, plotted on the x-axis, with the likelihood that that outcome could happen, plotted on the y-axis.

Probability distributions come in two flavors: discrete and continuous. A discrete probability distribution might represent the winnings on a lottery ticket paired with the likelihood of winning that amount. One uber-famous continuous probability distribution is the Normal Distribution.

On the far right of a probability distribution graph is the probability that, in three years, you sell your screenplay for $5,000,000. Then, far to the right of the middle is the probability you sell your screenplay in the next five years for $100,000. Just right of the middle is the probability you sell for $1...but to the shady guy at the Coffee Bean who’s promising you an 8-picture deal at Paramount. And just left of the middle is the probability you're still a barista.

The most likely stuff lives in the middle. As we slide towards either end, things get less and less likely.

So why is it called a distribution? Because the potential outcomes, i.e. things like winning a lottery, or selling a screenplay (they're kind of the same thing), or meeting someone on Tinder whose picture was less than 10 years and 20 pounds old...carries a range.

That is, the potential outcomes are distributed on a long line that then gets visually mapped to explain the character, or feeling, that describes this set of potentialities.

The most common continuous probability distribution is the Normal curve (or Normal distribution). You may know it better by its somewhat common nickname: the Bell curve. The mean (located in the middle, where the peak is) is usually labeled mu, which represents a population mean. The units on each side are plus and minus one, two, and three standard deviations. Sigma (no, uh...not Freud) is the symbol for a population standard deviation.

The Normal Curve was developed when researchers started comparing tons of measurements of things, like heights of giraffes, or diameters of plastic lids for drink cups, or lengths of…well, let’s just say it got a little competitive in the lab.

Turns out that tons of things, both man-made and nature-made, end up having a Normal curve shape to their measurements.

Example:

With heights of women, they found that a certain height (5 feet 4 inches) showed up more than any other. That one height showed up with the greatest probability. Heights taller than 5' 4" and heights shorter than 5' 4" showed up less often.

The farther the height was from 5' 4", the less likely it was to occur. Because, well...really tall women and really short women aren’t that common. And "average height" women are very common. When they plotted the heights and their associated probabilities (with thousands of results), they got a shape that became known as the Normal Curve.

Because of the shape of the Normal Curve, 68% of all the possible data lands between the first tick marks on each side of the mean (plus and minus one sigma). 95% of all the possible data lands between the second tick marks on each side of the mean (plus and minus two sigma) and 99.7% of all the possible data lands between the third tick marks on each side of the mean (plus and minus three sigma).

Graphically, the Empirical Rule shakes out like this. In the words of Master Yoda, "Worth memorizing, this curve is." We can use these percentages to determine how much of the possible data lands between different values on the Normal Curve. Let's say we get curious and decide to measure the length of the tail of every ring-tailed lemur we come across. Which, on the streets of Silicon Valley, is actually more than you’d think. Then we plot those tail lengths along with how often they showed up. We'd get a Normal Curve of tail lengths. The mean, or average, tail length would be at the peak in the middle, meaning that it was the measurement we got most often.

The tick marks on the x-axis would be found by adding the standard deviation of the tail lengths to the mean once, twice, and three times, and then subtracting the standard deviation from the mean once, twice, and three times. 68% of the lemurs we measured would have tail lengths between one sigma and negative one sigma. 95% of the lemurs we measured would have tail lengths between two sigma and negative two sigma. 99.7% of the lemurs we measured would have tail lengths between three sigma and negative three sigma.

The Empirical Rule isn’t the only game in town when it comes to Normal curving. There are other kinds of probability distributions that don’t cover every possible number, decimal, and fraction: discrete probability distributions. They usually hang out in tables, and sometimes in formulas.

Turning eighteen is great. You can vote. You can be drafted. You can buy lottery tickets. One quick scratch-off and you could on easy street, right? Eh, maybe not so easy. Grab a magnifying glass. Peep at the backside of the lottery ticket. There's a probability distribution on the back. It shows all the prizes you could win. It also shows the probabilities (or likelihood) that you win those prizes. It's a downer, though, so maybe you should ignore it and just scratch. You want that new Jacuzzi with shiatsu massaging jets, and it ain’t cheap.

This is a discrete probability distribution. Which just means we have a fixed number of outcomes. In this case, there are six possible outcomes. We can win five different dollar amounts and we can also "win" zilch. Check out the probability of "winning" $0. 78% of the time, you get nada. And there's a 1 in 2,000 (or 0.05%) chance of winning $100. It would have been better if Grandma had just given us the money she used to buy the ticket, instead of the ticket itself.

There are a few other kinds of discrete probability distributions. All of them can be placed in tables if we want. One specifically has a swaggy formula that helps us generate the probabilities for each possible outcome...and it's known as the Binomial Probability Distribution. BPD for short. To see this thing in action, we need to have a situation where there are only two things that can happen. We'll call winning any kind of moolah on that lottery ticket a "success." We'll call ending up with squat "failure." The BPD requires exactly two possible outcomes. If there are more than two, we can’t use the BPD.

The BPD also requires that the chance of a success always stays the same. If there's a 22% chance of winning on the first ticket of that kind, there needs to be a 22% chance of winning for all of those same kinds of tickets. The BPD also requires that we don't just spend the rest of our lives scratching off those tickets. We have to pick a set number of tickets we're gonna scratch and... stick to it.

The BPD would tell us that we have only a 3.75% chance of winning on half of those tickets. You'd have been better off spending the money on gas station nachos. At least then you'd have the stomachache to remember your money by.

Or maybe just toss your money in the trash and, uh... skip the middleman.