Probability without equally likely events

so far we've been dealing with one way of thinking about probability and that was the probability of of a occurring is the number of events that satisfy a number of events that satisfy a satisfy a overall of the equally likely events number of equally likely events oh and this is all of the equally likely all of the equally likely events and so in the case of a fair coin the probability of heads well it's a fair coin so there's two equally likely events and we're saying one of them satisfies being heads so there's a 1/2 chance of that you having a heads the same thing for tails if you took a if you took a die and you said the probability of getting an even number when you roll the die well there's six equally likely events and there's three even numbers you could get you could get two of four or six so there's three even numbers so once again you have a 1/2 chance of that happening and this is a really good model where you have equally likely events happening now I'm going to change things up a little bit so I'm going to draw a line here because this was just one way if they came up probability now we're going to introduce another one that's more helpful when we can't think about equally likely events and in particular I'm going to set up an unfair coin so this right over here is going to be my unfair coin so that is my coin well I could draw the coin so it's a gold coin this time it is unfair one side of that coin is a little heavier than the other so even though it's meant to look fair so it still has that picture of I know some president or something on on one side of it so this is the head side this is heads and then obviously on the back you have you have tails but as I mentioned this is an unfair coin and now I'm going to make an interesting statement about this unfair coin and one that really doesn't fit into the mold that I set up over here and this interesting statement is that we have more than a 50/50 chance of getting heads or more than a 50% chance or more than a 1/2 chance getting heads I'm going to say that the probability of getting heads for this coin right over here is 60% or another way to say it it's 0.6 or another way to say it it is 6 out of 10 or another way to say it it is 3/5 and this might make intuitive sense to you and hopefully it does a little bit but I want you to realize that this is fundamentally different than what we were saying before because now we can't say that there are two equally likely events there are two possible events you can either get heads or tails we're not assuming that the coin won't fall on its edge that that's impossible so you're either going to get heads and heads or tails but we can never we but they're not equally likely anymore so we really can't do this kind of counting the number of events satisfy something over all of the possible events in this situation in order to visualize the probability we have to kind of take what's kind of called a frequentist approach or think about it in terms of frequency pomp probability and the way to conceptualize a 60% chance of getting heads is to think if we had a super large number of trials if we were to just flip this coin this is saying that if we were to flip this coin a gazillion times we would expect that 60% of those would come up heads it's unclear how I determined that this is 60% maybe I ran a computer simulation maybe I know exactly all of the physics of this and I can completely model how it's going to fall every time or maybe it's it's I've actually just run a ton of trials I flipped the coin a million times and I said wow 60% of those 600,000 of those came up heads and then we can make a similar statement about tails so if the probability of heads is 60% the probability of tails well there's only two possibilities heads or tails so if I say the probability of heads or tails it's going to be equal to one because you're going to get one of those two things you have a hundred percent chance of getting a heads or a tails and these are mutually exclusive events you can't have both of them so this is going to be the probability of tails is going to be 100 percent 100 percent minus the probability of getting heads and this of course is 60 percent so it's 100 percent minus 60 percent or 40 percent or as a decimal 0.4 or as a fraction for tensor as a simplified fraction 2 over fifths so once again this probability is saying one thing we can't say that we can't say equally likely events we could say that if we're going to do a gazillion of these we would expect as we get more and more and more trials more and more flips 40% of those would be heads now with that out of the way let's actually do some problems with this so let's think about the probability of getting heads on our first flip and heads on our second flip so once again these are independent events the coin has no memory regardless of what I got on the first flip I have an equal chance of getting heads on the second flip it doesn't matter if I got heads or tails on the first so this is the probability of heads on the first flip times the probability of heads on the second flip and we already know probability of heads and on any flip is going to be 60 percent or another way to say it I'll write it as a decimal makes the math a little bit easier 0.6 0.6 and we can just multiply I'll do it right over here so this is 0.6 times 0.6 now it's always good to do a reality check if I'm take this is one one way to think about it is I'm taking six tenths of six tenths so it should be a little bit more than half of 6/10 so probably a little bit more than three tenths and then the and we've explained this in detail where we talked about multiplying decimals but we essentially just multiplied the numbers not thinking about the decimals at first 6 times 6 is 36 and then you count the number of digits we have to the right of the decimal we have one two to the right of the decimal so we're going to have two to the right of the decimal in our answer so it is 0.36 and that makes sense we're taking 60% of 0.6 we're taking 0.6 0.6 a little bit more than half of 0.6 and once again it's a little bit more than 0.3 so this also makes sense so it's 0.36 or another way to think about it is there's a 36% probability that we get two heads in a row given this unfair coin remember if it was a fair coin it would be 1/2 times 1/2 which is 1/4 which is 25% and it makes sense that this is more than that now let's think about a slightly more complicated example let's say the probability and now let's let's say the probability of getting a tails on the first flip getting a heads on the second flip and then getting a tails let me do this in a new color and then getting a tails on the third flip so this is going to be equal to the probability of getting a tails on the first flip because these are all independent events having one event if you know that has it you had a tail on the first flip that doesn't affect the probability of getting a heads on the second flip so times the probability of getting a heads on the second flip and then that's times the probability of getting a tails on the third flip times the probability of tails on the third flip and the probability of getting a tails on any flip we know is 0.4 the probability of getting a heads on any flip probability of heads on any flip is is 0.6 and then the probability of getting a tails on any flip is 0.4 is equal to 0.4 and so once again we can just multiply these so 0.4 times 0.6 so if we multiply if we multiply 0.4 times 0.6 there's actually a couple of ways we can think about it well we could literally say look we're multiplying four times six times four and then we have three numbers behind the decimal point so let's do it that way 4 times 6 is 24 24 24 times 4 is 96 so it gives us we get 9 we write a 9 6 but remember we have three numbers behind the decimal point so it's one one behind to the right of the decimal there one to the right of the decimal there one to the right of decimal there so three to the right so we need three to the right of the decimal in our answer so one two we need one more to the right of the decimal so our answer is zero point zero nine six or another way to think about it is write an equal sign here this is equal to a nine point six percent chance so there's a little bit less than a 10 percent chance or a little bit less than one in ten chance of when we flip this coin three times us getting exactly a tails on the first flip a heads on the second flip and a tails on the third flip