7. Suppose at this time of year in Auckland, there is a 50%...

7. Suppose at this time of year in Auckland, there is a 50% chance of rain. Given that it rains, there is an 80% chance the forecaster will be correct in predicting "Rain". Given that it does not rain, there is 70% chance the forecaster will be correct in predicting "No Rain". (a) The sample space for this problem is ? = {"rain" day and "no rain" day}. Define two events mentioned in the question. (b) Using the events you defined in (a), write down the information given in terms of probability statements. (c) What is the probability that a randomly chosen day is a rainy day and the weather forecast predicts that it rains? (d) If the weather forecast says it rains on a particular day, what is the probability that it actually rains? (e) What is the probability that a randomly chosen day is either a rainy day or the weather forecast predicts that it rains?

Transcript

00:01 So we have a situation in which we either have rain, or we have no rain, and they talk about that being the sample space, no rain.

00:09 And then we have predicting.

00:11 And we have predicting rain and not rain, predicting, well, let's go predicting, predicting, not rain.

00:24 And we have predicting rain.

00:28 I can get my symbolism right, predicting rain, and predicting not rain.

00:35 And we know that this probability is 0 .5 and this is 0 .5.

00:41 And given that, and they talk about, given that we have rain, probability of predicting rain was given as 0 .8.

01:00 So that will be up here and this would be predicting no rate.

01:03 And then we also have that idea that if, given that there is no rain, and the way they give it is the probability that they predict no rain is equal to 0 .7.

01:22 So this is 0 .7, and this is 0 .3.

01:27 And so let's move on to question c, and we want to find what is the probability of rain and they predicted rain.

01:38 And so that is multiplying these two branches of the tree, the 0 .5, times the point eight, which is point four.

01:48 On part d, we want to determine, given that they predicted rain, what's the likelihood that it's actually going to rain? and that means we have to look at all the different ways that they can predict rain, and then we want to find the probability of rain and predict rain, which we already know that answer, because that's what we just found in the previous problem.

02:16 And the probability of predicting well, there are two ways.

02:19 It rains and we predict rain, which is 0 .4.

02:24 Or you can have no rain and predict rain, which is 0 .5 times 0 .3, which is 0 .15.

02:32 So our answer will be 0 .4 divided by 0 .55, and as a fraction, that's going to end up being 40 out of 55.

02:44 And five goes into their eight times and five goes into their 11 times.

02:49 So we could say eight 11s.

02:51 If you want that as a decimal, eight divided by 11 will be a repeating decimal of 0 .72 repeating.

02:59 For part d, for part e, we have the probability of rain or the probability of predicting rain...

Question

Please give Ace some feedback