Video: EG19M2-Statistics-Q05

EG19M2-Statistics-Q05

03:07

Video Transcript

If 𝐴 and 𝐡 are two independent events from a sample space 𝑆 such that the probability of 𝐡 is 0.6 and the probability of 𝐴 or 𝐡, or we could say probability of 𝐴 union 𝐡, is equal to 0.68, find the probability of 𝐴.

It’s important to know that 𝐴 and 𝐡 are two independent events. This means we can use specific formulas to help us find the probability of 𝐴. For two independent events, 𝐴 and 𝐡, the probability of 𝐴 or 𝐡 is equal to the probability of 𝐴 plus the probability of 𝐡 minus the probability of 𝐴 and 𝐡. And we also have the formula that the probability of 𝐴 and 𝐡 is equal to the probability of 𝐴 times the probability of 𝐡. So we’re asked to find the probability of 𝐴. We’re given the probability of 𝐡. And we’re also given the probability of 𝐴 or 𝐡. And both of the given pieces of information are found in this first formula. So let’s use this one.

So let’s begin replacing items. We know that the probability of 𝐴 or 𝐡 is 0.68. We are also given that the probability of 𝐡 is 0.6. We do not know the probability of 𝐴 and 𝐡. And we’re trying to find the probability of 𝐴. So we need to plug in something for this probability of 𝐴 and 𝐡. Well, we have this other formula. The probability of 𝐴 and 𝐡 can be replaced with the probability of 𝐴 times the probability of 𝐡. And we know the probability of 𝐡; it’s 0.6. Now, instead of writing probability of 𝐴 times 0.6, it’s more commonly written as 0.6 times the probability of 𝐴.

Now, if we would rearrange these, where we have the probability of 𝐴 in the same spot but we switch the last two, so we have the negative 0.6 probability of 𝐴 and then plus 0.6. Notice both of these have a probability of 𝐴. So the first one that says probability of 𝐴 actually has a one in front. So we could take one times the probability of 𝐴 minus 0.6 times the probability of 𝐴. And one minus 0.6 would give us 0.4 times the probability of 𝐴. And then we bring down the 0.6.

We also could have looked at this another way. We could have taken out what these two terms had in common, probability of 𝐴. And if we would take 𝑃 of 𝐴, the probability of 𝐴, out, we would have one minus 0.6. And subtracting this, we get 0.4. So we will write this as 0.4 times the probability of 𝐴, just like we got.

So now we can actually solve for the probability of 𝐴. Let’s begin by subtracting 0.6 from both sides of the equation. And doing so, we get 0.08 is equal to 0.4 times the probability of 𝐴. Divide both sides by 0.4, and we find that the probability of 𝐴 is 0.2. So, once again, the probability of 𝐴 is equal to 0.2.

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