Back To CourseStatistics 101: Principles of Statistics
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Imagine walking into a casino for a game of dice, knowing exactly what to bet and how much you will win or lose. This is called the expected value of the game, and in this lesson we will learn how to calculate it.
In casinos, there is a game called 'craps,' which involves rolling two six-sided dice. The rules are somewhat complicated, but understanding the probabilities of the different combinations of the dice gives a person a solid advantage in the game.
Suppose you play a game where you roll two six-sided die. And, suppose the rules of the game are: you lose $3 if you get a sum of 2, 4, or 10. You lose $2 if you get a sum of 7, but win $1 for anything else. If you continue to play this game, what can you expect to win or lose in the long run?
So you may recall that the expected value, called E of X, of a situation is the sum of each value, called X, times its probability, called P of X. In mathematical notation this is written like this:
For this game, there are three events to consider:
And each of these occurrences has a value associated with them:
We can think of the amount won or lost as the values of the discrete random variable X. And so, X is the amount of money won or lost per roll of the dice. Plus, each of these values has a probability associated with it.
So the key to this is to know the probabilities of each event. For this, we need to examine what happens when two dice are rolled. Even though the die may look the same, from a mathematical perspective they are two distinct die. They are independent of each other. In probability, independence means they don't affect each other. So each die is independent of the other because one die doesn't affect the outcome of the other.
To illustrate, let's make one die blue and the other die red.
Each die has six sides, and so there are six possible outcomes for each die when rolled individually. However, when rolled together, there are 6 * 6 = 36 possible outcomes! This may be surprising, but the diagram below illustrates all the outcomes and all the sums when two dice are rolled.
So, for example, you could get a 1 on the red die and a 1 on the blue die, which adds to 2. And, actually, this is the only way to get a sum of 2. And so the probability of getting a sum of 2 when you roll two dice is 1 out of 36, which is about 0.028, or a 2.8% chance!
We can look at the table to find the probability of any of the sums for the two dice. So, for a sum of 4 the possibilities are 1 on red + 3 on blue, 2 on red + 2 on blue, or 3 on red + 1 on blue. The reason we do not count 2 + 2 twice is because 2 on red + 2 on blue is the same as 2 on blue + 2 on red!
Thus, the probability of a sum of 4 is 3 out of 36, which is about 0.083.
And the probability of a sum of 10 when you roll a dice is? You can pause the video here to calculate this from the table.
I hope you calculated 3/36. You can see from the table that there are three ways to get a sum of 10: 4 + 6, 5 + 5 or 6 + 4.
So let's go back to the game. You lose $3 if you get a sum of 2, 4 or 10. And now we know there is a 1/36 probability of getting a sum of 2, a 3/36 probability of getting a sum of 4 and also a 3/36 probability of getting a sum of 10. Since we are interested in all of these outcomes, we get a total probability of 1/36 + 3/36 + 3/36 = 7/36, or about a 0.194 probability of getting a sum of 2, 4 or a 10 when you roll two dice.
What about getting a sum of 7? What is its probability? You can pause the video here to see if you can calculate it.
So I hope you got 6/36, which is about 17%! So, we see there are 6 ways to get a sum of 7: 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, 6 + 1.
There are more ways to get a sum of 7 than any other sum, and it is sometimes called lucky seven! Although in this case, it is unlucky for you since you will lose $2 if you get it!
So how do you win this game? You need to roll anything else but the above sums of 2, 4, 10 or 7. And what is the probability of doing this? One of the properties of probabilities for a particular situation is that they all must add to 1. More formally: the sum of all the probabilities in a probability distribution add to 1.
A probability distribution lists all the outcomes of a particular situation and all the corresponding probabilities. In this case we are talking about the sum of the numbers on the two six-sided dice.
So, we already know that the probability of getting a sum of 2, 4 or 10 is 7/36 and the probability of getting a sum of 7 is 6/36. So what is the probability of getting some other sum?
Since they must all add to 1, we can just total the probabilities we know and subtract the result from 1! In other words the probability of getting a sum of anything else other than 2, 4, 10 or 7 is: 1 - 7/36 - 6/36 = 23/36, or approximately 0.64 or 64%. So there's a 64% chance you win something in this game, and that something is a dollar!
Here's the table from before with the probabilities:
What can you expect in this game if you play it many times?
To calculate the expected value we multiply the value of each event by its probability and then add the results. So, for the event of getting a sum of 2, 4 or 10, we multiply -3 times 7/36, which equals -21/36. And for the event of getting a sum of 7, we multiply -2 times 6/36, which equals -12/36. And finally, for the event of getting anything else, we multiply +1 times 23/36 which equals 23/36.
Adding them all together, we get an expected value of: -21/36 - 12/36 + 23/36 = -10/36 or -0.28. In other words, if you continued to play this game, your average winnings would be -0.28 per roll, or a loss of 28 cents per roll of the dice!
This is summarized in the table below:
And remember, your loss is the casino's gain. So if 100,000 people played this game over the course of a week, what would the casino make on this one game? Well, they would make 28 cents per person times 100,000 = $28,000 times the number of rolls each person played! This is not too shabby a return on investment for the casino!
We have seen that there are 36 possible outcomes when you roll two six-sided dice. Suppose you play a game using three six-sided dice. How many possible outcomes are there now?
Well, since the dice are independent of each other and each die has 6 sides, there are a total of 6 * 6 * 6 = 216 outcomes when you roll three dice! Now we will list all the outcomes here, but if you are ever feeling bored and need something to occupy time, try listing all the outcomes when rolling three dice. I can start you off 1 + 1 + 1, 1 + 1 + 2, 1 + 1 + 3, and so forth.
So let's extend this a little further. Suppose you play a game with three six-sided dice. The rules are: if you roll three sixes, which is a sum of 18, you win $100. But, if you roll anything else, you lose $1. What do you expect to win or lose in this game in the long run?
To answer this, we need to know the probability of getting three sixes when we roll three dice. Well, there is only one way to get three sixes when we roll three dice, and that is a six on the first die, a six on the second die and a six on the third die. And since there are a total of 216 outcomes, the probability of 3 sixes is 1/216!
So what is the probability of getting anything else? 1 - 1/216 = 215/216
To calculate the expected value, we multiply the value times it probability and sum the results. So the expected value of this game is: (100 * 1/216) + (-1 * 215/216) = -115/216 = -53 cents, approximately. So you can expect to lose about 53 cents on average for every roll of the dice! This is shown in the table below.
So far we have discussed playing with six-sided dice. But what if a die had a different amount of sides? Like a four-sided die or a 10-sided die? Well, the rules used to calculate the probabilities and expected values using these types of die are the same, although the probabilities will vary. For example, let's look at a four-sided die, which is called a tetrahedron, and let's suppose it is numbered 1-4 on its sides.
Since we are dealing with a four-sided die, when you roll it, the side we are interested in is the side that it sits on, not the side facing up. Why? Because a four-sided die cannot have a face sitting vertically upwards! Notice the number on the bottom side of the die labels the face that it is sitting on.
So, how many outcomes are there when we roll two four-sided die? Well, just like the six-sided die, they are independent of each other and so there are 4 * 4 = 16 possible outcomes. You can list them just as we did with the two six-sided dice, but note there can only be sums of 2 through 8.
So what is the probability of getting a sum of 7 when you roll two four-sided dice? To answer this, let's list the different ways to get a sum of 7 with two four-sided dice. To get a sum of 7 we can a 3 on the first die and a 4 on the second die or a 4 on the first die and a 3 on the second die. Thus, there are only two ways to get a sum of 7 when rolling two four-sided dice. And so, the probability of getting a sum of seven when rolling two four-sided dice is 2/16 or 1/8!
In this video we looked at the nitty-gritty behind rolling dice. We saw that with multiple die, each die is independent of each other. And so specifically, when you roll two six-sided dice, there are a total of 6 * 6 = 36 possible outcomes. But all of the outcomes result in sums from 2 through 12, with varying probabilities ranging from 1/36 for a sum of 2, to 6/36 for a sum of 7 and back down to 1/36 for two sixes, or a sum of 12.
We also saw that all the probabilities of the distribution added to one. And, we learned how to calculate an expected win or loss in a game of dice by multiplying each value by its probability and then adding the results. We also looked at variations on games of dice, and you saw that when you roll three six-sided dice, there are a total of 6 * 6 * 6 = 216 outcomes.
And when you roll two four-sided dice, there are a total of 4 * 4 = 16 possible outcomes. Listing the various outcomes allows us to calculate their probabilities, and knowing the probabilities allows us to calculate expected values. The expected value is really an average value of the game, be it a win or a loss.
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Back To CourseStatistics 101: Principles of Statistics
11 chapters | 144 lessons | 9 flashcard sets