### Video Transcript

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

Itβs important to know that π΄ and π΅ are two independent events. This means we can use specific formulas to help us find the probability of π΄. For two independent events, π΄ and π΅, the probability of π΄ or π΅ is equal to the probability of π΄ plus the probability of π΅ minus the probability of π΄ and π΅. And we also have the formula that the probability of π΄ and π΅ is equal to the probability of π΄ times the probability of π΅. So weβre asked to find the probability of π΄. Weβre given the probability of π΅. And weβre also given the probability of π΄ or π΅. And both of the given pieces of information are found in this first formula. So letβs use this one.

So letβs begin replacing items. We know that the probability of π΄ or π΅ is 0.68. We are also given that the probability of π΅ is 0.6. We do not know the probability of π΄ and π΅. And weβre trying to find the probability of π΄. So we need to plug in something for this probability of π΄ and π΅. Well, we have this other formula. The probability of π΄ and π΅ can be replaced with the probability of π΄ times the probability of π΅. And we know the probability of π΅; itβs 0.6. Now, instead of writing probability of π΄ times 0.6, itβs more commonly written as 0.6 times the probability of π΄.

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

So now we can actually solve for the probability of π΄. Letβs 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 π΄. Divide both sides by 0.4, and we find that the probability of π΄ is 0.2. So, once again, the probability of π΄ is equal to 0.2.