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# Dependent probability example

## Video transcript

let's do another one of these dependent probability problems you have four coins in a bag three of them are unfair and that they have a 45 percent chance of coming up tails when flipped the rest are fair so for the rest of them you have a 50% chance of tails or a 50% chance of heads you randomly choose one coin from the bag and flip it four times what is the percent probability of getting four heads so let's think about it when we put our hand in the bag and we take one of the coins out there's some probability that we get an unfair coin and three of the four coins are unfair so there's a three-fourths probability that we are we get an unfair coin an unfair coin and then there is only one out of the four coins that's fair so there was a 1/4 probability 1/4 probability that I get a fair coin now given that I have so unfair let's remind ourselves an unfair coin has a 45 percent chance of coming up tails so this means that I have a 45 percent chance of tails which also means we have to be careful here because they're asking us about heads if I have a 45 percent chance of getting tails that means I have a 55 percent chance of getting heads right whatever I have a 100 percent chance of getting one of these two if it's 45 percent for tails 100 minus 45 is fifty five for heads for the fair coin I have a fifty percent chance of tails and a fifty percent chance of heads 50 percent heads fair enough now I want to know in either of these scenarios what is a percent probability of getting four heads so if given I've got the unfair coin the probability of getting four heads is going to be fifty five percent for each of those flips so the probability of getting exactly four heads is going to be it's actually going to be 0.55 times 0.55 times zero point five five times 0.55 and so the probability of picking an unfair coin and getting four heads in a row is going to be equal to 3/4 times all of this business over here so that's 3/4 times and this is 0.55 times itself four times so I could write it as 0.55 to the fourth power and we'll get the calculator out in a second actually calculate what this is now let's do the same thing for the fair coin if I did pick a fair coin the probability of getting heads four times in a row is going to be 0.5 times 0.5 times 0.5 times 0.5 or the probability of getting the fair coin which is 1/4 probability 1/4 chance times the probability and getting 4 heads in a row is going to be 1/4 times all of this so it's going to be 1/4 times this is just 0.5 times itself 4 times so that's 0.5 to the fourth power so let's get the calculator out to calculate either one of these so we get 3/4 3/4 times and it knows that when I do the multiplication it's not in the denominator here so it's 3/4 it's times and I'll just do it in parentheses which I don't have to do in parentheses because it knows order of operations so 0.55 to the fourth power is equal to zero point let me write it down let me take it off the screen so I can write it down properly actually let me just do both of these calculations so this probability is that one right over there and then this one down here is 1/4 times 0.5 to the fourth power so it's equal to that right over there and so let's be clear the probability the probability of picking the unfair coin and then getting four heads in a row is this top number it's like six point roughly six point nine percent chance that you get the unfair coin and then get four heads in the row the probability that you get the fair coin and then get four heads in the road is even lower you it's only a 1.6 percent chance now the problem is getting four heads in a row I either way is going to be the sum of this and this or the sum of that and that which is going to be let me keep my calculator out so it's going to be equal to I can just take the previous ends let me just retype it so I don't confuse you so point zero one five six two five plus point zero six eight six two nine where's that two nine six eight seven five six eight seven five I'm going to round it anyway so it won't matter too much so if I take the sum let me take this off screen and then let me so I can still see it and then let me write it so what I got here this one is zero point zero six eight six two nine and I'll round at seven and this down here this down here was zero point zero one five six two five and when you add these two up because we just care about getting four heads either way this is the probability of getting it this way with the unfair coin this is the probability of getting it with the fair coin we want it either way so let's add the two which we already did in our calculator so if you add that number to that number you get zero point zero eight four two five and it keeps going but I'm just going to round it so this is the same thing as this is equal to eight point four two five percent if I want to round it a little bit more eight point four three percent chance of getting four heads in a row and once again that should be a that's a slightly higher number then if all of the coins were fair because there's a there's a three fourth chance that I get a coin that has a better than even chance of getting head so that's why this number is going to be a little bit higher than the probability if I had a fair coin of just getting four heads in a row