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you have eight coins in a bag three of them are unfair in that they have a 60% chance of coming up heads when flipped the rest are fair coin so if three are unfair the rest are eight coins when I say that or when this problem says that they are fair coins it means that they have a 50/50 chance of coming up either heads or tails you randomly choose one coin from the bag and flip it two times what is the percent probability of getting two heads so this is an interesting question but if we break it down essentially with the decision tree it'll help break it down a little bit better so let's say that so we have a bag three of them are unfair so that we can even visualize the bag you don't have to do this all the time so we have out of the fair coins and white one two three four four five fair coins and we have three unfair coins one two three and this whole thing is my bag right over here that is my bag of coins if I were to when I take my hand in if I were to take any of these white coins there's a 50% chance that it gets heads on any flip the odds of getting two heads in a row would be 50% times 50% for these white coins but I don't know I'm going to get a white coin if I get the one of these orange coins I have a 60% chance of coming up heads and the odds of coming up 60% if I if I have picked one of these orange coins the probability of getting heads twice is going to be 60 percent times 60% so how do I factor in this idea that I don't know if I've picked a white fair coin or an orange unfair coin we will assume that the coins actually aren't white Norns they all look like regular coins so what I'll do is I'll draw a little bit of 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 pick a fair coin that I pick a fair coin and there is some probability that I pick an unfair coin unfair coin and so what is the probability that I pick a fair coin well 1 2 3 4 5 out of the total 8 coins are fair so there is a 5/8 probability there is a 5/8 I'll write here on the branch actually so there's a 5/8 chance that I pick a fair coin and then there is a 3 1 2 3 out of 8 chance that I pick an unfair coin so if I were to just tell you what's the probability of picking a fair coin you'd say a 5/8 what's the probability of an unfair coin 3/8 and you could convert that to a decimal or a percentage or whatever you'd like now given that I have picked a fair coin what is the probability that I that I will get heads twice so let me write it this way this is just notation right here so the probability of I'll call it heads heads so I get two heads in a row given given that I have a fair coin given that I have a fair coin it looks like very fancy notation but just like look if you knew for a fact that that coin you had is absolutely fair that it has a 50% chance of coming up heads what is the probability of getting two heads in a row well then we can just say well that's just going to be 50% so 50% times 50% times 50% which is equal to 25% which is equal to 25% now what is the probability that you so if you want if you want to know what is the probability that you picked a fair coin and you got two heads in a row so given that you have a fair coin it's a 25% chance that you have two heads in a row but the probability of picking a fair coin and then given the fair coin getting two heads in a row will be the 5/8 times the 25% so the probability so this whole branch I should maybe draw it this way the probability of this whole series events of happening so starting with you picking the fair coin and then getting two heads in a row will be I'll write it this way it will be 5/8 5 over 8 times this right over here times the 0.25 times 0.25 I want to make it very clear the 0.25 probability of getting two heads in a row given that you knew that you had to get a you got a fair coin but the probability of this whole series of events happening you would have to multiply this times the probability that you actually got a fair coin so another way of thinking about it is this is the probability this is the probability that you got a fair coin that you got a fair coin and then you have two heads in a row and that you have two heads in a row now let's do the same thing for the unfair coin so the probability I'll do that in the same green color the probability that I get heads heads given that my coin is unfair given that is unfair so if you were to somehow know that your coin is unfair what is the probability of getting two heads in a row well in this unfair coin it has a 60% chance of coming up heads so it will be equal to 0.6 times zero point six which is 0.36 so if you have an unfair coin if you know for a fact that you have an unfair coin if that is a given you have a 36 percent chance of getting two heads in a row now if you want to know the probability of this whole series events the probability that you picked an unfair coin and and you get two heads in a row so the probability of unfair unfair and two heads in a row given that you had that unfair coin you would multiply this 3/8 times the 0.36 so this will be equal to this will be equal to the 3/8 this 3/8 times times 0.36 and so let's get a calculator out and calculate these so if I take 5/8 5/8 times 0.25 I get 15-point or I'll just write it as a decimal point one five six two five so this is equal to zero point one five six two five and then if I do the other part so if I have 3/8 times 0.36 that gives me Oh point 1 3 5 so this is zero point 1 3 5 so if someone were to ask you what's the probability of getting a halving of picking the fair coin and then getting head two heads in a row with that fair coin you would get this number if some want to say what's the probability of you pick the unfair coin and then get two heads in a row with that unfair coin you would get this number now if someone were to say either way either way what's the probability of getting two heads in a row because that's what they're asking us here what is the probability of getting two heads so we could get it through this method by chance picking the fair coin or through this method by chance picking the unfair coin so since we can do it either way we can sum up the probabilities either of these events either of these events meet our constraints so we can just add these two things up so let's do that so we can add 0.135 plus 0.15 6 2 5 gives us point 2 9 1 2 5 so point 0 point 2 9 1 to 5 that's when we add 0.15 6 to 5 plus 0 point 1 3 5 will equal this and 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 29 point one two five percent or if we were to round to the nearest hundredths then this would be well this would be the exact number or we could say it's approximately 29 point one three percent depending on how much we need to round it so we have a little less than a third chance of this happening and the reason why remember if it was a fair coin there was only B if everything in the bag was a fair coin there'd be a 25% chance of this happening because you would just say okay any of these they're all the same flip it twice 25% chance our chance is a little bit higher because there's some probability there's a a chance that we pick a coin that has a higher than even chance of coming up heads