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suppose that Erica simultaneously rolls a six-sided die and a four-sided die let a be the event that she rolls doubles let me write this a be the event that she rolls doubles and B be the event that the four-sided die is a for use the sample space of possible outcomes below to answer each of the following questions fair enough what is probability of a the probability that Erica rolls doubles well over here we have our sample space of possible outcomes each of these are equally likely and so let's see how many of them there are there are 1 2 3 4 by 1 2 3 4 5 6 so there are 24 possible outcomes which makes sense there's 4 possible outcomes for the four-sided die and six possible outcomes for the six-sided die so you have a total of 24 equally likely outcomes so probability of let me write it here so probability of a probability of a is going to be the fraction of the 24 equally likely outcomes that involve event a that she rolls doubles so let's think about that this is she has rolled doubles 1 and a 1 they don't look the same but they're both ones let's see we have a 2 and a 2 we have a 3 and a 3 and we have a 4 and a 4 and it's impossible to have a 5 and a 5 because the four-sided die only goes up to 4 so there's four possibilities for of the 24 equally likely possibilities involve rolling doubles so there's a 4 24th probability or if we divide the numerator and denominator by 4 it is a 1/6 probability that Erica a 1/6 probability that Erica rolls doubles what is probability of B the probability that the four-sided die is a 4 so the probability of B well once again there's 24 equally likely possibilities and how many of them involve the four-sided die being a 4 you have all of these right over here involve a four-sided die being a four so this is one two three four five six of the 24 equally likely possibilities or you could say one-fourth of the equally likely possibilities or the probability is the probability is one-fourth which makes sense because probability of beat it kind of ignores the six-sided die and it just says well what's the probability that the four sided dice four well that's one out of the four possible outcomes for that four sided die what is the probability of a given B the probability that Erica rolls doubles given that the four sided die is a four so let's just think about this a little bit probability of a of a given that B has happened given that B has happened so essentially we are restricting our equally likely possibilities now to the situation where B has happened given B means we're assuming that B has happened so now we're restricting our sample space of possible outcomes where B has happened to this right over here so now there are 1 2 3 4 5 6 equally like six equally likely outcomes and how many of them involve a happening well this one right over here that we had already we had already circled this is the one out of the six equally likely outcomes that involve that involve doubles so there's a 1/6 probability now that makes sense let me just write this down this is 1 over 6 why does this make sense because we the four-sided die we're assuming is a 4 so it's essentially this is analogous to saying when you roll a 6-sided die what's the probability that you get a 4 as well because that's the only way you're going to get a get doubles given that the four-sided die is 4 and we see that right over here the six-sided die has to be a 4 as well in order for this to be double because we're assuming where it's given that B we're given a bavette B we're restricting our sample space with event B what is the probability of B given a the probability the force iodized forgiven that erica rolls doubles so let's just think about that a little bit so the probability probability of B given a B given given that a is true so what's this going to be well we've already restrict our sample space to the to the essentially four equally out likely outcomes that that a has happened so where a is true I guess I could say so there's one two three four and how many of them involve event B being true well the only one the only one of these four that involve event B being true is this one right over here where we've got our doubles so there is a one one-fourth probability that if we assume it given that we've gotten doubles the probability that the four sided die is a four so this is a one-fourth probability and that makes that make sense if we've got doubles and one of them is a four sided die we either have doubles at one doubles at two doubles at three or doubles at four you see that here doubles one doubles two doubles three doubles four well given that what's the probability that the four sided die is four well that means that's the one one out of these four outcomes where it's double for us right over here all right what is the probability of a and B the probability that Erika rolls doubles and the second die is four so this means if both a and B happened well let's look at this let me write it here let me do it in this we did a new color so the probability of all right and here in a neutral color probability of a and B probability of a and B is equal to well now we're looking at once again we have 24 equally likely outcomes we have 24 equally likely outcomes and how many of them involve a and B well to get a and B you have to have doubles and and the 4-sided die needs to be a four essentially you have to have doubles four well there's only one out come out of the 24 equally likely outcomes that meets that that meets that situation this one right over here so there is a 1/4 one sorry one twenty-fourth probability so 1 over 24 what is probability of a get the times probability of B given a well here we could just go back to our numbers right over here probability of a that's going to be 1 over 6 let me do that in magenta color I like to keep my colors be careful about my colors that's 1/6 times probability of B given a so probability of B given a is 1/4 right over here times 1/4 which is curious enough 124 124 what is probability of B times probability of a given B well probability of B we figured out is 1/4 1/4 and the probability of a given B is 1/6 times 1/6 times 1/6 which is equal to which is equal to 124 now does it make sense that the probability of a and B is 124 the probability of a times probability of B given a is 124 and the probability of B times probability of a given B they're all 124 is this always going to be the case well sure think about what probability of a and B means well it means that they both happened but that's the same way is saying well what's the probability what's the probability of let's just say a is happening well now for B and a to happen it's just going to be that times the probability that B is true given that a is true because we're you could kind of say well we're already kind of constraining it we're already multiplying by the probability of a being true and now we're multiplying by the probability that B is true given a is true actually often like to swap these around just it gets a little bit clearer in my head so this one let's just write it like this the probability of B given a time's the probability of a so this is the probability that event a is true and this is the probability that B event B is true given that we know that a is true and it completely makes sense that this is going to be the same thing as the probability of a and B clearly this is the probability of both of these both a and B happening you can go the other way around the probability of a given B times the probability of B that would also be so B we're saying B needs to be true and that given that B is true that a needs to be true as well so it makes complete sense that this is going to be the probability of a and B as well