Video Transcript
The Venn diagram shows the
probabilities of events 𝐴 and 𝐵 occurring or not occurring in different
combinations. Now, there are four parts to this
question. Calculate the value of 𝑥. Hence, calculate the probability of
𝐴. Calculate the probability of 𝐴
given 𝐵. And finally, are 𝐴 and 𝐵
independent events?
So what we’re gonna do is start
with this first part, which is calculate the value of 𝑥. Well, what we know about a Venn
diagram like this is that when we’re looking at probability, all the values must add
together to equal one. So, therefore, we have one-seventh
plus four-sevenths plus two over 21 plus 𝑥 equals one.
So now if we want to add our
fractions, what we want to do is have a common denominator. So we can convert one-seventh to
three over 21 and four-sevenths to 12 over 21. So then we’ve got three over 21
plus 12 over 21 plus two over 21 plus 𝑥 equals one. So, therefore, we’ve got 17 over 21
plus 𝑥 equals one. Well, to make one, what we need is
21 over 21. So, therefore, 𝑥 must be equal to
four over 21.
Okay, great, so now what we can do
is move on to the second part of the question. Hence, calculate the probability of
𝐴. Well, the probability of 𝐴 is
given by anything inside the oval that we have here, which is 𝐴. So we’ve got one-seventh and four
over 21. So, therefore, the probability of
𝐴 is gonna be equal to a seventh plus four over 21. Well, again, what we need is a
common denominator. So we’re gonna make 21 our common
denominator. And we’ve already seen that
one-seventh is equal to three over 21. So, therefore, we can say the
probability of 𝐴 is equal to three over 21 plus four over 21. So then if we add our numerators,
we’re gonna get the probability of 𝐴 is equal to seven over 21.
Okay, great, have we finished
there? Well, no, because what we can do is
we can actually cancel down or simplify our fraction. And we can do this by dividing both
the numerator and denominator by seven. So, when we do that, we get a
third. So, therefore, we can say that the
probability of 𝐴 is a third.
So now what we can do is move on to
the next part of the question, where we’re asked to calculate the probability of 𝐴
given 𝐵. So what this means is the
probability of event 𝐴 occurring given that event 𝐵 occurs. So what this question asks is
conditional probability, because we’re asked to calculate the probability of event
𝐴 occurring given that event 𝐵 occurs. So, if we look at the Venn diagram,
what we’ve got here is in pink of circle where event 𝐵 occurs. And then what we want to know is
the probability that 𝐴 occurs, which I’ve shaded in here orange, given these
conditions.
Well, we’ve got a formula to help
us solve this part of the problem because we know that the probability of 𝐴 given
𝐵 is equal to the probability of 𝐴 intersection 𝐵 divided by the probability of
𝐵. Well, we already know the
probability of 𝐴 intersection 𝐵 because this was 𝑥 in the first part of the
question, because what this means is the probability of 𝐴 and 𝐵. And we can work out the probability
of 𝐵 because this is equal to four over seven plus four over 21, which is gonna be
equal to 12 over 21 plus four over 21, which is equal to 16 over 21.
So now we have both parts of our
formula because we know that the probability of 𝐴 intersection 𝐵 is four over 21
and the probability of 𝐵 is 16 over 21. So, therefore, we know the
probability of 𝐴 given 𝐵 is four over 21 divided by 16 over 21. Well, if we remember how we divide
fractions, then what we do is we keep the first fraction the same. We change the divide to multiply,
and then we flip the second fraction or find the reciprocal of the second
fraction. So what we’ve got is four over 21
multiplied by 21 over 16.
So then what we could do is just
multiply the numerators and denominators. But to make it easier, what we can
do is a bit of simplifying first. And first of all, we can divide
both the numerators and denominators by 21 and then divide the numerator and
denominators by four. And what we’re left with is one
multiplied by one over one multiplied by four. So, therefore, we can say that the
probability of 𝐴 given 𝐵 is gonna be equal to a quarter.
Okay, great, so now what we can do
is move on to the final part of the question. Are 𝐴 and 𝐵 independent
events? Well, what we know is that two
events are independent if the probability of 𝐴 and 𝐵, or the probability of 𝐴
intersection 𝐵, is equal to the probability of 𝐴 multiplied by the probability of
𝐵.
Well, we’ve already calculated the
probability of 𝐴 and the probability of 𝐵 earlier on in our problem. So, therefore, what we can do is
multiply these together. So, when we do that, what we’re
gonna get is one over three multiplied by 16 over 21, which is gonna give us 16 over
63.
Well, now, what we want to do is
compare this to the probability of 𝐴 intersection 𝐵. Well, we know that the probability
of 𝐴 intersection 𝐵 is four over 21, because we worked this out in the first part
of the question. Well, we can convert four over 21
into something over 63, and that’s gonna be 12 over 63. And we do that by multiplying the
numerator and denominator by three. So, therefore, if we compare the
probability of 𝐴 multiplied by the probability of 𝐵 and the probability of 𝐴
intersection 𝐵, we can see that 16 over 63 is not equal to 12 over 63. So, therefore, we can say that
events 𝐴 and 𝐵 are not independent.
So, therefore, for a quick recap,
we can say that the value of 𝑥 is four over 21, the probability of 𝐴 is a third,
the probability of 𝐴 given 𝐵 is one-quarter, and no, events 𝐴 and 𝐵 are not
independent.
