# Question Video: Two-Way Tables, Conditional Probability, and the Relationship between Categorical Variables

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 1, 2, and 7 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.

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### Video Transcript

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.