"At least one" probability with coin flipping

now let's start to do some more interesting problems and one of these things that you'll find in probability is that you can always do a more interesting problem so now I'm going to think about I'm going to take a fair coin I'm going to flip it flip it three times and I want to find the probability of at least one head at least at least one at least one head out of the three flips so the easiest way to think about this is how many equally likely possibilities there are in the last video we saw if we flip three if we flip a coin three times there's eight possibilities for the first flip there's two possibilities second flip there's two possibilities and then the third flip there are two possibilities so 2 times 2 times 2 there are eight equally likely possibilities if I'm flipping a coin three times now how many of those possibilities have at least one head well we drew all of the possibilities over here so we just have to count how many of these have at least one head so that's one two three four five six seven so seven of these have at least one head in them and this last one does not so seven of the eight have at least one head now you're probably thinking okay Sal you were able to do it by writing out all of the possibilities but that would be really hard if I said at least one head out of you know 20 flips this worked well because I only had three flips so let me make it clear this is in 3 flips in in 3 in 3 flips this would have been a lot harder to do a more time-consuming to do if I had 20 flips is there some shortcut here is there's some other way to think about it and you couldn't you couldn't just do it in some simple way you can't just say oh probability of heads times probability of heads because if you've got probability if you've got heads the first time then you know you don't have to get heads that anymore or you could get heads again you don't have to so it becomes a little bit more a little bit more complicated but there is an easy way to think about it where you could use this methodology right over here you'll actually see this on a lot of exams where they make it seem like a harder problem but if you if you just think about it in the right way all of a sudden it becomes simpler one way to think about it is the probability of at least one heads in 3 foot is the same thing this is the same thing as the probability of not of not getting not getting all tails all tails right if we got all tails then we don't have at least one head so this these two things are equivalent the probability of getting at least one head in three flips is the same thing as the probability of not getting all tails in three flips let me write in three in three flips so what's the probability of not getting all tails well that's going to be one minus the probability of getting all tails the probability of getting all tails and since it's three flips it's the probability of tails tails and Tails because any of the other situations are going to have at least one head in them and that's all of the other possibilities and then this is the only other leftover possibility if you add them together you're going to get one let me let me write it this way the probability let me write it in a new color just so you see where this is coming from the probability of not all tails not all tails plus the probability of all tails probability of all tails well this is essentially exhaustive this is all of the possible circumstances so your chances of getting either not all tails or all tails and these are mutually exclusive so we can add them so the probability of let me write it this way the probability of the probability of not all tails or just to be clear what we're doing the probability of not all tails or the probability or the probability of all tails all tails is going to be equal to one these are mutually exclusive you're either going to have not all tails you're which means the head shows up or you're going to have all tails but you can't have both of these things happening in system mutually exclusive and you're saying the probability of this or this happening you can add their probabilities and this takes this is essentially all of the possible events so all of the probability of so this is essentially if you if you combine these this is the probability of any of the it's happening and that's going to be a one or a hundred percent chance so another way to think about it is the probability of not all tails is going to be one minus the probability of all tails so that's what we did right over here and the probability of all tails is pretty straightforward that's the probability of get it's going to be one half because you have a one half chance of getting a tails on the first flip times let me write it here a little clearer so this is going to be one minus the probability of getting all tails you will you have a one half chance of getting tails on the first flip and then you're going to have to get another tails on the second flip and then you're going to have to get another tails on the third flip and then 1/2 times 1/2 times 1/2 this is going to be this is going to be 1/8 and then 1 minus 1/8 or 8/8 minus 1/8 is going to be equal to is going to be equal to 7/8 so we can apply that to a problem that is harder to do than writing all of the scenarios like we did in the first problem we can say we can say probability the probability let's say we have 10 flips the probability of at least 1 head at least 1 head in 10 flips in 10 flips well we use the same idea this is the same thing as going this is going to be the product this is going to be equal to the probability of not not all tails not all tails in 10 flips not all tails so we're just saying the probability of not getting all of the flips going to be tail all of the flips is tails not all tails in 10 flips and this is going to be this is going to be 1 minus the probability of flipping tails 10 times so it's 1 minus 10 tails in a row 10 tails in a row and so this is going to be equal to this part right over here let me write this so this is going to be this one let me just rewrite it this is equal to 1 minus and this part is going to be well one tail another tail those 1/2 times 1/2 I'm going to do this 10 times I'm going to do this 10 times let me write this a little neater because I need a 1/2 so that's 5 6 7 8 9 and 10 and so we really just have to the numerator is going to be 1 so this is going to be 1 this is going to be equal to 1 let me do it in that same color of green this is going to be equal to 1 minus our numerator you just have 1 times itself 10 times so that's 1 and then on the botton denominator you have 2 times 2 is 4 4 times 2 is 8 16 32 64 128 256 512 1024 over 1024 this is the same exact same thing as 1 is 1024 over 1024 minus 1 minus 1 over 1024 over 1024 which is equal to 10 23 or 1023 1023 over 1024 we have a common denominator here so 1,000 doing that same blue over 1000 and 1024 so if you flip a coin 10 times in a row a fair coin your probability of getting at least one heads in that 10 flipped it's pretty high it's 1,000 23 over 1,024 you can get a calculator out to figure that out in terms of a percentage actually let me just do that just for fun so if we have 1,000 23 divided by 1024 that gives us you have a 99.9% chance that you're going to have at least at least one heads so this is if we round this is equal to 99.9% chance and I rounded a little bit it's actually slightly even slightly higher than that and this is a pretty powerful tool or a pretty powerful way to think about it because it would have taken you forever to write all of the scenarios down in fact there would have been 1020 four scenarios to write down so this would have this this doing this exercise for 10 flips would have would have would have taken up all of our time but when you think about in a slightly different way when you just say look the probability of getting at least one heads in 10 flips is the same thing as the probability of not getting all tails and that's 1 minus the probability of getting all tails and this is actually a pretty easy thing to think about