Simple hypothesis testing

let's say that we have four siblings right over here and they're trying to decide how to pick who should do the dishes each night and so the oldest sibling right over here he decides well look I'll just put all of our names into a into a bowl and then I'll just randomly pick one of our names out of the bowl each night and then that person is going to be so this is the bowl right over here and I'm just going to put four sheets of paper in there each of them it's going to have one of their names and then he's just going to randomly pick it out each night and then that's the person who's going to do their dishes so they all say well you know that's that seems like a reasonable reasonably fair thing to do and so they start that process so let's say that after the first three nights that he the the oldest the oldest brother here and I maybe let's call him let's call him Bill let's say after three nights bill has not had to do the dishes so at that point the rest of siblings are starting to think maybe just maybe something fishy is happening so what I want to think about is what is the probability of that happening what's the probability of three nights in a row bill does not get picked if we assume that we were randomly taking if bill was truly randomly taking these things out of the bowl and and not cheating in some way what's the probability that that would happen that three nights in a row bill would not be picked I encourage you to pause the video and think about that well let's think about the probability that Bill's not picked on a given night if it's truly random so we're going to assume we're going to assume that Bill's not cheating so assume assume truly random truly random and that each of the sheets of paper have a one in four chance of being picked what's the probability that bill does not get picked well there's so let me the probability that nut I guess I'm right this bill not picked on a night on a night well there's four equally likely outcomes and three of them result in bill not getting picked so there's a three-fourths probability that bill is not picked on a given night well what's the probability that Bill's not picked it three nights in a row let me write that down so the probability bill not picked three nights nights in a row well that's the probabilities he's not picked on the first night times the probability that he's not picked on the second night times the probability that he's not picked on the third night so that's going to be three to the third power or 3 times 3 times 3 so that's 27 over 4 to the 3rd power 4 times 4 times 4 is 64 and if we want to express that as a decimal so that is 27 I get my calculator out that is 27 divided by 64 is equal to and I'll just round to the nearest hundredth right here 0.42 so that is equal to 0.42 and so this doesn't seem that unlikely it's it's a little less likely than kind of even odds but it's not you know you wouldn't you wouldn't question someone's credibility if you know there's there's a 42 percent roughly a 42 percent chance that three nights in a row bill would not be picked so this this seems like if you're assuming truly random that that it's a reasonable your hypothesis that it's truly random seems seem you know there's a good chance that that you're right it's a 40 there's a 42 percent chance you would have the outcome you saw if your assumption is true but let's say let's say you keep doing this and you trust your older brother you know why why would he want to cheat out his his younger siblings but let's say that Bill's not picked 12 nights in a row so then everyone's starting to get everyone's starting to get a little bit everyone's starting to get a little bit suspicious suspicious with Bill with Bill right over here and so they say well you know well we're going to give him the benefit of doubt assuming that that he's being completely honest that this is completely random process what is the açaí that he would not be picked 12 nights in a row well just just write that down so the probability bill and it's really the same stuff that I just wrote up here I'll just read bill not picked 12 nights in a row well that's going to be three you're going to take 12 3 4 it's and multiply them together it's going to be 3/4 to the 12th power and what is this going to be equal to well let's see if you take well 3/4 says I'll just write 3 divided by 3/4 which is going to be 0.75 to the 12th power now this is a much smaller this is now if we actually this is going to be 0.3 I guess we could go to it we could go to one more decimal place here 0.32 or we could say so I mean it's this is zero point zero point zero three two I should say which is equal to so this is approximately equal to let me write that which is equal to three point two percent so now you have every right to start thinking that something is is getting fishy you could say well look you know if there was and this is what statisticians actually do they often just define a threshold and hey you know if the probability of this happening purely by chance is is more than 5% then I'll say well maybe it was happening by chance but if the probability of this happening purely by chance was way no and and this is the threshold of statisticians often use is 5% but that's somewhat arbitrarily defined but this is a fairly low probability that it would happen fairly by chance so you might be tempted to reject the hypothesis to reject the hypothesis that it was truly random that bill is cheating in some way and you could imagine if it wasn't 12 in a row was 20 in a row then this probability becomes really really really really really small and and so your hypothesis that it's truly random starts to really come and come into doubt