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Current time:0:00Total duration:4:56

one rainy Saturday morning out of woke Adam woke up to hear his mom complaining about the house being dirty mom is always grouchy when it rains Adams brother said to him so Adam decided to figure out if the statement was actually true for the next year he charted every time it rained and every time his mom was grouchy what he found was very interesting rainy days and his mom being grouchy were entirely independent events some of his data are shown in the table below fill in the missing values from the frequency table and let's see we have day this raining days not raining days and the total days that he kept data for and then he tagged laid on her say the raining day weather his mom was grouchy or not grouchy and on a not raining day whether his mom was grouchy or not grouchy there was a total of 35 days it rained 330 days that it didn't rain and then 7373 times his mom was grouchy and 292 times his mom was not grouchy so the first thing we said well how do we how do we figure this out we have these four boxes here it's not clear that we can we can just just we have enough information to fill it out just with this table but we have to remember what they told us they told us that his mom being grouchy and it raining were entirely independent independent events another way of saying that is the probability of his let me just in a color that you're more likely to see another way of saying that so independent events that means that the probability my pin is acting up a little bit probability that mom is grouchy so let me write that mom my pen is really mom mom is grouchy given it is raining given it is raining should it shouldn't really matter whether it's raining it should just be the same thing as the probability of mom of mom being grouchy grouchy in general it grouchy in general so what does that tell us well we can figure out the probability that mom is grouchy in general she is grouchy 73 out of 365 days so the probability that mom is grouchy in general is going to be is going to be 75 or 73 73 divided by 365 365 and so at least just based on the data we have that's the best estimate that mom is grouchy the probability that mom is grouchy it's the percentage of days that she's been grouchy so that is 0.2 so based on the based on the data the best estimate of the probability of mom being grouchy is 0.2 or or 20% and so we should have the probability of mom being grouchy given that it's range should be 20% as well so this number so given that it's raining we should also have 20% of the time mom is grouchy because these are independent events it shouldn't matter whether it's raining or not this should be 20% this should be this should be she should be 20 should be grouchy 20% of the time that it's raining and she should be grouchy 20% of the time that it's not rating that what would be consistent with the data saying that these were entirely independent events so what is 20% of 35 well 20% is 1/5 1/5 of 35 is 7 and once again all I did is I said 20% of 35 is 7 and if that's 7 then 35 minus 7 that's going to be 28 right over there and then if this is 7 and then 73 minus 7 is going to be 66 66 and 333 I guess there's a couple of ways we could do it we could take actually we could just take 292 minus 28 is going to be let's see 282 minus 8 would be 284 or minus other 2264 264 do the numbers all add up yeah 66 plus 264 is 330 so the key realization here is what he's saying he found was very interesting rainy days and his mom being grouchy were entirely independent of that means that the probability of his mom being grouchy should it shouldn't matter whether it's raining or not it should just be it should be the same probability of whether it's raining or not and our best estimate of the probability of his mom being grouchy is on the total days as 20% and so if the data is backing up that it's independent events then the best way to fill this out would be his mom being the probability of his mom being grouchy on a rainy day or not rainy day should be the same and that's what we filled out right over here