Video: Conditional Probability: Two-Way Tables

In this video, we will learn how to deal with the concept of conditional probability using joint frequencies presented in two-way tables.

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Video Transcript

In this video, we’ll learn how to deal with the concept of conditional probability using joint frequencies presented in two-way tables.

But what do we mean by conditional probability? It’s defined as the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome. For example, event A is, it’s raining outside, and event B is that your train is delayed. In conditional probability, we look at these events together and ask questions like, what is the probability that your train will be delayed given that it’s raining outside? We’d write this as shown, and this vertical line means given that.

One way we have to recall the outcomes of such events is in a two-way table. It’s a table that shows the frequency of two variables. We need to be able to complete two-way tables and calculate probabilities from them. So let’s see what this might look like.

A fanzine website for the TV show A Maze in Space collects data on the number of new alien species encountered by two starships on each season of the show. The data for seasons one, two, and seven are shown in the table below, split between the two starships Zeta and Geoda. Find the probability that a new alien species chosen at random was encountered by starship Geoda. Give your answer to three decimal places.

We want to find the probability that a new alien species chosen at random was encountered by starship Geoda. This doesn’t specify which season we’re interested in, so we’re going to use the total. And so we begin by recalling that the probability of an event occurring is found by dividing the number of ways that event can occur by the total number of possible outcomes. So in this case, we’re going to find the number of alien species found in total by starship Geoda. And then we’ll divide that by the total number of alien species encountered altogether.

Looking at the totals, we see that starship Geoda discovered a total of 72 alien species. And the total was 83. So the probability that a new alien species chosen at random was encountered by starship Geoda is found by dividing 72 by 83, which is 0.8674 and so on. Correct to three decimal places, that’s 0.867.

So that’s how we calculate a simple probability from a two-way table. Let’s now have a look at how to find conditional probabilities from these tables.

Given that a new alien species was encountered in season seven, find the probability that they were encountered by starship Geoda. Give your answer to three decimal places.

The phrase “given that” is an indication that we’re going to be calculating conditional probability. If we let A be the event that the alien species were encountered by starship Geoda and B be the event that this happened in season seven. Then we use this vertical line to show “given that.” We’re finding the probability that A occurs given that B has occurred. Now, the phrase “given that” essentially narrows down the criteria. We’re told that the alien species was encountered in season seven. So we’re only interested in these three pieces of data. Out of this list, we see that the number of alien species encountered by starship Geoda is eight.

Of course, the probability is found by dividing the number of ways that event can occur by the total number of outcomes. And the total number here is 13. So the probability that A happens given that B has happened is eight divided by 13, which is 0.61538 and so on. Correct to three decimal places, we see that given that a new alien species was encountered in season seven, the probability that they were encountered by starship Geoda is 0.615.

Let’s have a look at another example of finding probabilities from two-way tables.

The table below contains data from a survey of core gamers who were asked whether their preferred gaming platform was the smartphone, the console, or the PC. The gamers are split by gender. Find the probability that a core gamer chosen at random prefers using a console. Give your answer to three decimal places. Given that a core gamer prefers to play using a console, find the probability that they are male. Give your answer to three decimal places.

Now, firstly, we recall that if we’re trying to find the probability of an event occurring, we divide the number of ways that event can occur by the total number of outcomes. And the first part of this question asks us to find the probability that a gamer chosen at random prefers to use a console. Now, they don’t specify whether we’re interested in male or female gamers. So, in fact, we’re going to calculate the totals.

We begin by calculating the total number of gamers who prefer to use a smartphone. That’s 52 plus 48, which is 100. Similarly, to calculate the total number of gamers who preferred the console, we add 37 and 23, to give us 60. Finally, the total number of gamers who prefer to use the PC is 48 plus 35, which is 83. The total number of gamers questioned is found by adding all of the values in this column. That’s 100 plus 60 plus 83, which is 243.

Now, remember, we’re looking to find the probability that the gamer chosen at random prefers to use a console. So that’s this second row. The total number of outcomes or the total number of gamers here we calculated to be 243. So the probability that a core gamer chosen at random prefers to use a console is 60 divided by 243, which is 0.2469 and so on. That’s 0.247.

The second part of this question states that given that a core gamer prefers to play using a console, find the probability that they are male. This phrase “given that” is an indication that we’re going to use conditional probability. If we let event A be the event that the gamer chosen is male and event B be the event that they prefer to use a console, we use the bar notation to show that we’re trying to find the probability of A occurring given that B has occurred. And what this does is narrow down the data somewhat.

We’re told that the gamer prefers to play using a console. So we can narrow our data down into just those people who prefer to play using a console. And we want to find the probability that they are male. So we’re going to divide the number of male gamers who said they preferred using a console by the total number of gamers who said they preferred using a console. That’s 37 divided by 60. That’s 0.61666 and so on, which correct to three decimal places is 0.617. So the probability that a core gamer is male given that they prefer to play using a console is 0.617.

In our next example, we’ll quote and learn how to use a conditional probability formula.

Daniel and Jennifer are running for the presidency of the Students’ Union at their school. The votes they received from each of three classes are shown in the table. What is the probability that a student voted for Jennifer given that they are in the class B?

Remember, the phrase “given that” indicates that we’re working with conditional probability. We’ll say that event A is that a student voted for Jennifer. And event B is that the student is in class B. Then this vertical line means given that. We’re finding the probability that A occurs given that B has already occurred. And one way we can do this is to narrow down the table based on the information that we’ve been given.

We’re told that that student is in class B, so we narrow it down to everyone in class B. In this case, we’re interested in the number of students that voted for Jennifer. That’s 195. And remember, to find the probability of an event occurring, we divide the number of ways that event can occur by the total number of outcomes. Here, the total number of possible outcomes is the total number of students in class B. That’s 169 plus 195, which is 364. And so the probability that a student voted for Jennifer given that they’re in class B is 195 over 364, which simplifies to 15 over 28.

Now, in fact, this is a perfectly valid method for answering this question. But there is a formula we can use. We say that to find the probability of A given B, we divide the probability of A intersection B ⁠— in other words, A and B ⁠— by the probability of B. So in this case, what’s A intersection B? Well, A was the number of students who voted for Jennifer and B was the number of students in class B. We’re looking for the intersection, the students that voted for Jennifer and are in class B. There are 195 of them.

The probability of choosing one of these at random is found by dividing 195 by the total number of students asked altogether. That’s 507 plus 494. That gives us a total of 1001. So the probability of A intersection B, in other words, the probability that a student voted for Jennifer and are in class B, is 195 out of 1001. And what about the probability of B, in other words, the probability that they’re in class B? Well, we already saw that there are 364 students in class B. So the probability of B occurring is 364 divided by 1001. And so, if we apply the formula, we get 195 over 1001 divided by 364 over 1001. Notice that this gives us the exact same answer as earlier, 195 over 364, which simplifies to 15 over 28.

In our final example, we’ll look at how we can use information from two-way tables to help us decide where the two events are independent.

Data is collected from the TV show A Maze in Space on the number of new alien species first contact is made with. The data for starship Zeta in seasons one, two, and seven are shown in the table below. The data have also been categorized by whether the crew member who made first contact was male or female. From the table, find the probability that first contact was made with a new alien species by a female crew member. Give your answer to three decimal places.

We want to find the probability that first contact was made by a female crew member. Let’s call that 𝑃 of F, where F is the event that a female crew member was the person who made first contact. And we know that to find the probability of an event occurring, we divide the number of ways that event can occur by the total number of outcomes. The information on female crew members is this second row. And we know there are a total of 37 first contacts made by female crew members.

The total number of outcomes are the total numbers of first contacts made with a new alien species; that’s 72. So the probability then that first contact was made with a new alien species by a female crew member is 37 divided by 72. That’s 0.5138 and so on, which correct to three decimal places is 0.514.

We’re now going to move on to the second part of this question.

The second part of this question says, find the probability that first contact was made in season one and by a female crew member. Give your answer to three decimal places.

This time, not only are we interested in first contact being made by a female crew member, but this must occur in season one. If F is the event that the crew member is female and S one is the event that first contact was made in season one, we want to find the probability of S one intersection F. Remember, that just means S one and F. So let’s begin by finding the number of first contacts made in season one by a female crew member. Well, that’s 16. The total number of first contacts made is still 72. So the probability the first contact is made in season one and by a female crew member is 16 divided by 72. That’s 0.2 recurring, which is 0.222 correct to three decimal places.

Let’s now have a look at the third part of this question.

Given that first contact was made with an alien species chosen at random, in season one, find the probability that first contact was made by a female crew member. Give your answer to three decimal places.

This phrase “given that” is really useful because it tells us that we’re working with conditional probability. And we can narrow down our data set. We’re told the first contact was made with an alien species chosen at random from season one. And so we narrow the data down to just the results from season one. We use this vertical line to represent “given that.” And we see that we want to find the probability that first contact was made by a female crew member given that it happened in season one.

Well, in season one, 16 crew members made first contact. That’s out of a total of 28. So the probability that this happens then is 16 divided by 28, which is 0.5714 and so on. Correct to three decimal places, that’s 0.571.

We’ll now consider the fourth and final part of this question.

Are the events S one which is first contact made in season one and female independent?

Remember, two events are independent if one occurring doesn’t affect the probability of the other occurring. And whilst we could probably try and use a bit of common sense, there are some formulae we can use. The first is for two events A and B. And this says that if these events are independent, then the probability of A intersection B is equal to the probability of A times the probability of B. In other words, the probability of A and B will be equal to the product of their two respective probabilities.

Now, alternatively, we can say that if two events A and B are independent, then the probability of A occurring given that B has occurred must be equal to the probability of A occurring. And so we can say that if our events are independent, then the probability that the crew member is female given that first contact is made in season one will be equal to the probability that they are female. So let’s see if this is true.

We worked out that the probability that they’re female given that first contact was made in season one is 0.571. And we worked out the probability of them being female in general was 0.514. Well, these are not equal. And so we can say no, these events are not independent. And similarly, we could’ve used the alternative formula. We calculated the probability that they were female. And the first contact was made in season one was 0.222.

We calculated the probability of them being female to be 0.514. And we could calculate the probability that first contact was made in season one. It would be 28 divided by 72; that’s 0.389. Now, in fact, when we find the product of 𝑃 of F and 𝑃 of S one, we get 0.199. That’s not equal to 0.222. So that’s an alternative way we could show that these events are not independent.

In this video, we’ve learned that in a two-way table, we organize the frequencies for the categories of two categorical variables. We saw that we can calculate conditional probabilities by reading directly from a two-way table. And the use of the phrase “given that” is an indication that we can narrow our results down. Finally, we saw that for two events A and B, we can determine whether they are independent if the probability of A given B is equal to the probability of A. If this is not true, that’s an indication that A and B are dependent events.

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