Video: Using Probabilities in a Venn Diagram to Decide Whether Events Are Independent

In a sample space 𝑆, the probabilities are shown for the combination of events occurring. Are 𝐴 and 𝐡 independent events?

03:36

Video Transcript

In a sample space 𝑆, the probabilities are shown for the combination of events occurring. Are 𝐴 and 𝐡 independent events?

Firstly, let’s recall the conditions for two events 𝐴 and 𝐡 to be independent. 𝐴 and 𝐡 are independent if the probability of 𝐴 intersect 𝐡 is equal to the probability of 𝐴 timesed by the probability of 𝐡. Now let’s look at what each of these terms in this equation means.

Let’s first look at how we can find the probability of 𝐴 intersect 𝐡. This is the probability of 𝐴 and 𝐡 both occurring. Let’s show which region on the Venn diagram is represented by this probability.

Now the region which represents 𝐴 intersect 𝐡 is the region where 𝐴 and 𝐡 overlap. So that’s this region here, which I’ve shaded in. Now we can look at the Venn diagram from the question. And we see in this region that the probability is five 19ths. And so the probability of 𝐴 intersect 𝐡 is equal to five 19ths.

Next, let’s look at how we find the probability of event 𝐴 occurring. The probability of event 𝐴 occurring is represented by the sum of all the probabilities in the circle representing 𝐴. And this circle is this one shaded here. And we mustn’t forget that it also includes this region here where it also overlaps 𝐡.

So now to find the probability of 𝐴, we have to add together all of the probabilities in this region. From the Venn diagram given in the question, we can see that the probabilities are four 19ths and five 19ths. So this gives us the probability of 𝐴 is equal to four 19ths plus five 19ths, which is the same as nine 19ths.

Now let’s look at how to find the probability of 𝐡. The probability of 𝐡 occurring is represented by all the probabilities inside the region represented by 𝐡. This region is the region shaded here. Again, we mustn’t forget to include the bit where overlaps 𝐴, here.

Now to find the probability of 𝐡 occurring, we need to look back at our Venn diagram from the question and add together all of the probabilities in this region. So we see that the probabilities are five 19ths and five 19ths. And so we get that the probability of 𝐡 is equal to five 19ths plus five 19ths, which is just 10 19ths.

So now we have found the probability of 𝐴 intersect 𝐡, the probability of 𝐴, and the probability of 𝐡. Now all that is left to do is to take the probability of 𝐴 and multiply it by the probability of 𝐡 and see if this is equal to the probability of 𝐴 intersect 𝐡.

The probability of 𝐴 times the probability of 𝐡 is nine 19ths times 10 19ths. And this gives us 190 over 361. Now we have the probability of 𝐴 intersect 𝐡 is equal to five 19ths. And this is not equal to 190 over 361. So we come to the conclusion that 𝐴 and 𝐡 are not independent.

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