### 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.