Understand how probability shapes everyday decisions and games of chance. Learn practical methods to calculate probability using examples from coin tosses to games like craps.
Key Insights
- Calculate probabilities practically by using Excel's RAND function to generate random numbers between 0 and 1, facilitating dynamic and repeatable probability testing.
- Understand the law of large numbers, which explains that outcomes approach theoretical probabilities as trials increase; flipping a coin five times might yield 60% heads, but 500 flips approach the true probability of 50%.
- Analyze the probabilities involved in games of chance, such as drawing a king from a 52-card deck (7.69% probability) and the complex probabilities in the dice game craps, which reveal a 49.3% overall chance of winning, giving the house a slight edge.
Note: These materials offer prospective students a preview of how our classes are structured. Students enrolled in this course will receive access to the full set of materials, including video lectures, project-based assignments, and instructor feedback.
Probability one. We're going to use probability to determine the likelihood that an event will or will not occur. So a couple of facts about probability.
The probability of an event that cannot occur is zero. The probability of an event that cannot occur is zero. The probability of an event that must occur is one.
So you're looking at either zero or one. It's very similar to correlation or the R coefficient. Every probability is between zero and one.
You can think of this as percentages between zero percent and 100 percent. One is standing for 100 percent. The greater the number of trials, the more accurate the probability will be.
And you must remember that nothing is 100 percent certain. That's why it's called probability and not certainty. There is a function in Excel that we can use to determine probability randomly.
And that function is the RAND function. You simply type equal RAND, open and close parentheses. And when you press ENTER, since this is a volatile function, you will get a different result each time.
If you run this in your own spreadsheet, you will get a different answer than I get. There is never going to be the same result that we both get because this is a random function. When I press ENTER, I have 0.79. That is equivalent to 79 percent.
Later on in the course, we will use the RAND function to determine some random probability of an event occurring. Now, if you want to get a new number, there's a couple of things you could do. You can simply double click on another cell and then press ENTER.
Even though you're not currently in that cell, that value will automatically update. If you don't want to double click on a cell so you can get a new result, you can head over to the formulas tab. And from the formulas tab, there's a button specifically for recalculating all the formulas and functions in your spreadsheet.
If I click calculate now, I get a new result. So that's 50 percent. I'll click it again.
15 percent and so on. Now let's talk about the law of large numbers. A fair coin flipped five times may give you heads 60 percent of the time.
So that means if you get heads three out of five times, that is 60 percent. But if you flip it, if you flip a coin 500 times, it will almost certainly be exactly 50 percent. So this is basically saying that you need to have enough trials to reach your theoretical number.
Flipping a coin 50 percent of the time you should get heads and 50 percent of the times you get tails. But if you flip it only five times, that's an odd number. So you'll have to conduct more trials.
If you flip a coin twice and you get heads twice, that does not mean that you're more likely to get heads than tails. So empirically or experimentally, we think the probability of getting heads at first is 60 percent if we get that result three out of five times. But later, you realize it's almost 50 percent.
Theoretically, the probability is 50 percent. The more trials we run, the closer to the probability we will be. Now, let's take a look at a couple of other probabilities.
Probabilities come up when you're playing games, because games, in most cases, are games of chance. So for exercise one, I want to determine the probability of drawing a king from a fair deck of 52 playing cards without jokers. Now, this is a fixed, predetermined probability.
A fair deck means that I do not have five kings or three kings. I have exactly four. And I have a total of 52 cards, not including jokers.
So what is the probability that when I draw a card from the deck of 52 cards that I will draw a king? Well, that's four out of 52. If I write the formula for that, it's going to be equal to four divided by 52. When I press ENTER, my percentage is 7.69 percent.
So this lets me know what I'm playing with in terms of my probability of drawing a king. Here's another exercise. Determine the probability of grabbing a yellow prize from a basket of prizes.
This probability can change if the prizes change. So what is the probability of me drawing a yellow prize out of a total of 10? Well, it depends on how many yellow prizes I have. Now, what's fair is that all the values for the items I have in the bag should add up to 10.
So it's simply going to be equal the number of the item divided by the total. And that's how you calculate a percentage. So I'm going to go over here and select the total.
Now, since I want to use the total multiple times, I'll press F4 to lock the position of the total. I'll press ENTER. And so the probability of me drawing a red toy is 50 percent.
Why? Because there are five red toys out of 10, and that's 50 percent. I'll auto fill this down. The probability of pulling out a yellow toy is 30 percent, green toy 20 percent.
So that's basically the probability as it relates to the yellow prize. Let's take a look at probabilities here. Again, a fair game is where everything adds, all the values add up to 100 percent.
So all the variables involved add up to 100 percent. So I want to prove that the sum of probabilities rule for a six-sided, I want to prove the sum of probabilities rule for a six-sided die. So I'm calculating the probability of rolling either a one to five in terms of values for a dice or rolling a six.
A one to five is going to give me an 83 percent probability because out of the five values, there are five chances out of six that I could roll one to five. What are the odds of me rolling a six? Well, it's one out of six. So that's 17 percent.
So I want to make sure that these odds are on the up and up. So I'm going to go to 83 percent. I'm going to add 17 percent.
When I press ENTER, I get 100 percent. So that is a fair game. If I had returned either something like 90 percent or 110 percent, then there's something off.
All right, so now we'll move on to exercise four and we're going to talk about the game of craps. So the game of craps is played with two dice. Each dice can roll a number from one to six.
Now, we've determined the probability of you returning certain combinations. We have all the combinations listed here and they show up as a sideways pyramid. So you have the most combinations for the number seven.
And number 11, you just have two. And then there are some numbers where you'll only have one chance of rolling that number, like a two and a 12. So a two is one and one.
Twelve is six and six. All right, so we can see the probabilities for each number. And we've calculated them here.
That work is already done for you. So what we do need to do at the moment is explain how to play craps. And then we'll take a look at our probabilities of winning.
So how do you play craps? Well, in the game of craps, you win if you roll a seven or 11 on your first roll. So what are the odds of you rolling a seven or 11? Well, the odds for rolling a seven are 16.67 percent. The odds of rolling an 11 are 5.56. If you add those together, you have a 22 percent chance.
So some people may think that is a chance I'm willing to take. And I know nothing is certain, but 22 percent I think is pretty good. That's what someone might think.
Now, how would you lose the game? Well, you would lose if you roll a two or a three or a one. That is automatically craps. Those are the values in red.
Those are the values you definitely don't want to roll. Now, what are the chances of you rolling any of those individual numbers? 11 percent. So this is another reason why you think the odds are in your favor.
But you may forget that if you avoid a two or three or a one and you don't roll a seven or 11, well, then any other number that you roll, you have a 66.67 percent of getting those numbers. So what happens if you roll any other number, any other number other than two, three and one, you'll get something called the point. Now, the way it works is when you get the point, before you can continue on with the game, you have to roll the same exact number again in order for you to continue.
Now your odds have gone down because if you must roll that same number again, we have a listing of all the numbers that are not two, three or one or seven or 11. Your odds of rolling that same number again are lower, lower than rolling a seven or 11. So you cannot roll a seven or 11 before you roll that same number twice.
And you definitely don't want to get two, three or one or you'll lose. Now the odds start to not look as good. So before you roll, there's a higher chance of getting a seven or a good number twice.
So I'm going to select the percentages there. So you have 16% seven chance of rolling, getting a seven, then a good number twice. So it's kind of dicey.
So what are the odds of you not rolling a seven? 83.33%. Now, the goal of this is not to teach you how to play craps. The goal is to look at your probabilities when it comes to playing this game. And it may look promising at first, but statistically, after looking at all the rules for the game and the percentages, the odds of winning at craps is 49.3%. So the house edges out the player just a little bit, and that's enough for the house to make a profit.
As they say, the house always wins. We have a link that you can click on that will explain the entire game from start to finish. So this section has been focused on looking at the probabilities of playing certain games and decisions that you can make within those games based on the probabilities.
There are additional exercises that are very similar to the ones we already looked at. You can do those on your own. This is the first example of probability.
After this, we're going to take another look at probability when it comes to selecting items from a bag with or without replacement. So that's coming up. Thank you for watching.