Video: Calculating the Union of Two Events Involving Conditional Probabilities

Suppose that 𝐴 and 𝐡 are events with probabilities 𝑃(𝐴) = 0.34 and 𝑃(𝐡) = 0.52. Given that 𝑃(𝐡 | 𝐴) = 0.615, find 𝑃(𝐴 βˆͺ 𝐡).

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Video Transcript

Suppose that 𝐴 and 𝐡 are events with probabilities. The probability of 𝐴 is equal to 0.34 and the probability of 𝐡 is equal to 0.52. Given that the probability of 𝐡 given 𝐴 is equal to 0.615, find the probability of 𝐴 union 𝐡.

The first thing I wanted to go through was just a bit of notation that we’ve used. The first bit is this vertical line between the 𝐡 and the 𝐴. What this means is 𝐡 given that 𝐴 occurs. So we’re saying the probability that 𝐡 occurs given that 𝐴 occurs. Then, we have this other bit of notation, which is U shape between 𝐴 and 𝐡. What this means is 𝐴 union 𝐡. What we mean by union is 𝐴 or 𝐡. So what’s the probability of 𝐴 or 𝐡 occurring.

Now, to find the probability of 𝐴 union 𝐡, what we need to use is a couple of rules that we know. Well, the first rule is the rule of multiplication. And what this tells us is that the probability of 𝐴 and 𝐡 or 𝐴 intersection 𝐡 is equal to the probability of 𝐴 multiplied by the probability of 𝐡 given that 𝐴 has occurred. And the other rule that we can take a look at is the rule of addition. And this rule tells us that the probability of 𝐴 union 𝐡 is equal to the probability of 𝐴 plus the probability of 𝐡 minus the probability of 𝐴 intersection of 𝐴 and 𝐡.

Well, this is great because this gives us exactly what we want, because we want to find the probability of 𝐴 union 𝐡. However, there’s one problem. We’re not given in the question the value of the probability of 𝐴 intersection 𝐡. So what are we gonna do? Well, what we can do is we can combine both of our rules. So what we can do to combine our rules is substitute in the probability of 𝐴 multiplied by the probability of 𝐡 given 𝐴 for our value for the probability of 𝐴 intersection 𝐡. And that’s because we can substitute that in from our rule of multiplication.

So therefore, our combined rule is gonna be the probability of 𝐴 union 𝐡 is equal to the probability of 𝐴 plus the probability of 𝐡 minus the probability of 𝐴 multiplied by the probability of 𝐡 given 𝐴. Well, we have all these values because all of these values are values we’ve been given in the question. So the last thing we need to do now is substitute these in to find out what the probability of 𝐴 union 𝐡 is. So when we substitute in our values, we get that the probability of 𝐴 union 𝐡 is gonna be equal to 0.34 plus 0.52 minus 0.34 multiplied by 0.615.

So therefore, this is gonna give us a final answer for the probability of 𝐴 union 𝐡 of 0.6509.

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