Hovedindhold

## Videregående statistik og sandsynlighed

### Enhed 5: Lektion 6

Multiplikationsregel for afhængige hændelser# Afhængig sandsynlighed: mønter

Vi overvejer, hvordan sandsynligheden for et udfald kan afhænge af et andet udfald i det her eksempel. Lavet af Sal Khan.

## Video udskrift

you have a coins in the bag three of them are unfair that they have a 60% chance and coming ahead when to lift the rest are five three are unfair the rest are eight when I say that or when this problems isn't their fares of coming up either you randomly choose one coin from what is the percent probability of getting to an interesting question but if we break it down decision tree it'll help break it down a little bit better so let's say that so we have a bag and three of them are unfair bag you don't have to do this all the time so we have the fair coin white one two three four or five fair family I've said we have three unfair coins this whole thing is my bag right over here is my bagon point if I were to funding my hand in if I were take any of these white coins there's a 50% chance I guess head odds of getting two heads in a row would be fifty percent to two percent over in these white times but I don't know I'm going to get a white person if I get the one of these origins I have a 60% chance of coming up if I have picked one of these orange coins the probability of getting heads twice so how do I faster in this idea that I don't know if I've picked within this white coin or an orange unfair Owen will assume that they all look like regular coins so what I'll use a little bit of a decision tree here I guess maybe I could call it a probability tree so there's some probability that I can pick a fair coin then since you knew the point before the I think a fair coin since you and there's some probability that I pick an unfair coin unfair coin in so what is the probability that I think since being fair coin well one two three four five out of the portal may in coin are fair so there is a finite probability dean next e-scooter there is a 5/8 all right here on the branch actually so there's a 5/8 chance and I pick a fair coin foon and then there is a three one two three out of eight yes and then pick an unfair coin so far just tell you what's wrong you're picking a 5/8 what's the probability of an unfair create that in decimal or whatever you'd like so since you now own given that I have a fair coin you mind what is the probability that I that I will get heads twice then for me right is the way cool this is just notation right here so the probability I call heads heads for schools or I can do heads in a row given its are given that I have a fair chance we need info given that I have a fair coin it looks like very fancy notation just like if you knew for a fact that that coin you had absolutely fair that has a 50% chance of heads what is the probability of getting two two in a row well we can just say well that's just going to be 50% it's less percent so 50% times 50% times 50% to 25% yeah God is equal to 25 percent or two percent now what is the probability since you that's a fault you find so if you want if I want to know what is the probability the fair coin and you got two heads in a row so given that name out of a fair coin at the 29 position similar to theirs in row where the probability of picking a fair coin and then given the fair coin getting two heads in a row will be the 5/8 so the probability this whole branch the probability of this is happening starting with you pick on the fair coin and then getting two heads in a row by hand will be all right in this way will be 5/8 pin I've over eight times this right over here times the point into five times the 0.25 I want to make it very clear disease wastin five is the probability of getting two heads in a row coma given the probability this whole series of events happening you would have to multi scale first times probably got a fair pose could become so another way if they come out it is this is the probability this is the probability that you got a fair coin that you got it this morning and it's retired you have two heads in a row and that you have two heads in a row or the unfair coin would fit so the probability I'll do that in the same green color that my coin is unfair given that is unfair so if you were to somehow know that when point is unfair what is the probability of getting two heads in a row well the unfair coin restrictions just a chance of coming up heads won't fit so it will be equal to zero that is a given 36 percent chance of getting to know the probability in this whole series events the probability that you would fit and and get into a probability of unfair unfair given that you had that unfair coin yeah you would multiply this 3 times the point that was given to me so let's get a calculator out and calculate these there's no mine so if I take 5/8 pin pi divided by income a few more times 0.25 let's go to some of this means I get 15 point all right I'll just write it as a decimal point one five six five six - this is equal to zero point one five six - five and then the partners if I have three take you today by eight nude time 0.036 that gives me point one three five this is zero point no one company five so someone were to ask you what's the probability of getting a having picking the fair coin and then getting head to heads in a row with that fair coin you would get this no dignity if someone say what's the process you're hopefully on a fair point and then get two heads in a row that unfair coin they even get this number now if someone were to say nicer way so either way heads in a row because that's what they're asking us here through this method you could buy a chance picking the fair swing cons or through this method by chance the unfair coin so since we can do it either way we can sum up the probabilities either every event he had to either of these events um meet our constraints so we can dig it ahead in these two things own so let in so we can add 0.135 plus 0.15 6 into 6 5 gives us beyond I want to try it too thin so point zero point two nine one two five that's when we add zero point one five six two five plus zero point five if we want to write it as a percentage you essentially just multiply this times 100 and add the percentage sign there so this is equal to twenty nine point one 25% round for these hundreds then this you can exact number two or we could say it's approximately twenty nine point three percent depending on how much basically do for fans we have a little less than a third the reason why they're mean 25 percent chance of this happening because you would just say okay therefore any of these except they're all the same flip it twice these money finals and it's our chances a little higher because there's some probability there's a 3/8 chance that we pick a coin that has a higher than even chance of coming up heads