### Video Transcript

The probability of a day being rainy is 0.24, stormy 0.36, and rainy and stormy 0.14. Find the probability of it, one, being either rainy or stormy and, two, being rainy where it is not stormy.

We can use a Venn diagram to sort out these probabilities. We have the probability that it’s rainy, the probability that it’s stormy. The intersection of these two circles is the probability that it is both raining and storming. And we’re actually given that value. The probability that it’s raining and storming is 0.14. We’re also given that the probability that it is rainy is 0.24. The pink area in the Venn diagram is all of the probability that it’s raining.

We know that the probability that it’s raining and storming is 0.14. To find the probability that it’s raining but not storming, we would subtract 0.14 from 0.24. And we would get 0.10. We know that the probability of it raining equals the probability that it’s just raining plus the probability that it’s raining and storming. When we add those together, we get 0.24 which is what we were given. We need to follow the same process to find out what goes in this part of the Venn diagram, the probability that represents the time that it is stormy but not rainy.

We know the total probability of it being stormy is 0.36. So we take that total probability 0.36 and subtract the probability that it is both stormy and rainy. And we get 0.22. The probability that it is stormy but not rainy is 0.22. We know the total probability that it is storming equals the probability that it is stormy but not rainy, 0.22, plus the probability that it is stormy and rainy, 0.14. And when we add those together, we get 0.36 which is what we started with.

Using this diagram, we can answer both of those scenarios. The first one is the probability that it is rainy or stormy. Our first instinct would be to add 0.24 and 0.36, adding the probability that it’s rainy or the probability that it is stormy. But there’s an error here. Remember, that the probability that it is rainy is made up of two parts, the part that it’s raining and the part that it’s raining and storming. The stormy probability is also made up of two pieces, 0.24 and 0.14.

If you add 0.24 plus 0.36, you’re adding the probability that it is both raining and storming twice. We only need to consider the probability that it’s raining and storming once. We can correct this in one of two ways. We can add 0.24 plus 0.36 and then subtract 0.14. In this case, we would be adding the probability of rain plus the probability of storm and subtracting one of the probabilities that it’s raining and storming.

The other way we could do this would be to add each other probabilities from the Venn diagram, 0.10 plus 0.14 plus 0.22. This would be the probability that it’s rainy but not stormy, the probability that it’s both, and then the probability that it’s stormy but not rainy. Adding all three of these together, we get 0.46 which is the same thing we get when we add 0.24 plus 0.36 and then subtract 0.14. The probability that it’s either rainy or stormy is 0.46. Part two is asking the place where it is rainy and not stormy. On our Venn diagram, the place where it is rainy and not stormy is here. That probability is 0.10. And we can simplify that to say 0.1.