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Studying for a test? Prepare with these 8 lessons on Probability and combinatorics.
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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 coins. So if three are unfair, the rest are fair coins. 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 a decision tree, it'll help break it down a little bit better. So let's say that we have a bag, three of them are unfair. So we could even visualize a bag. You don't have to do this all the time. I'll do the fair coins in white. One, two, three, 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. 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 one of these orange coins, I have a 60% chance of coming up heads. If I have picked one of these orange coins, the probability of getting heads twice is going to be 60% 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'll assume that the coins actually aren't white and orange. They all look like regular coins. So what I'll do is I'll draw 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. And there's some probability that I pick an unfair coin. And so what is the probability that I pick a fair coin? Well, one, two, three, four, five out of the total eight coins are fair, so there is a 5/8 probability. I'll write it here on the branch, actually. So there's a 5/8 chance that I pick a fair coin, and then there is a 3-- one, two, three, 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 oh, 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 will get heads twice? So let me write it this way. Now 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 that I have a fair coin-- it looks like very fancy notation, but it's just like look, if you knew for a fact 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? Then we can just say, well, that's just going to be 50%, so 50% times 50%, which is equal to 25%. Which is equal to 25%. 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 this whole branch-- I should maybe draw it this way-- the probability of this whole series of events 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 times this right over here, times the 0.25. I want to make it very clear. The 0.25 is the probability of getting two heads in a row given that you knew that 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 that you got a fair coin 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. 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 0.6, which is 0.36. 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% chance of getting two heads in a row. Now if you want to know the probability of this whole series of events-- the probability that you picked an unfair coin and you get two heads in a row, so the probability of 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 3/8 times 0.36. And so let's get a calculator out and calculate these. So if I take 5 divided by 8 times 0.25, I get 0.15625. So this is equal to 0.15625. And then if I do the other part, so if I have 3 divided by 8 times 0.36, that gives me 0.135. So this is 0.135. So if someone were to ask you, what's the probability of picking the fair coin and then getting two heads in a row with that fair coin, you would get this number. If someone were to say, what's the probability of you picked 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, 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 meet our constraints. So we can just add these two things up. So let's do that. So we can add 0.135 plus 0.15625 gives us 0.29125. So point 0.29125, that's when we add 0.15625 plus 0.135 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 29.125%. Or if we were to round to the nearest hundredths, then this would be the exact number, or we could say it's approximately 29.13%, depending on how much we need to round it. So we have a little less than a 1/3 chance of this happening. And the reason why-- if everything in the bag was a fair coin, there'd be a 25% chance of this happening because you just say, OK 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 3/8 chance-- that we pick a coin that has a higher than even chance of coming up heads.