Day 7 HW   Conditional Probability + Independent vs Dependent Events

Day 7 HW Conditional Probability + Independent vs Dependent Events



in this lesson we will review a few of the probability concepts that we've learned over the last few lessons we'll talk about conditional probability independent events and dependent events look here a few of the formulas that we've learned dealing with probability one thing to keep in your mind is if we're dealing with a or B you're usually going to wind up adding whereas if you're trying to find the probability of a and B then you're going to be multiplying and then you have conditional probability all right the probability of a given B is equal to the intersection of a and B divided by the probability of the given part notice that it has to be the given a right given B we're dividing by be back on this first formula notice that basically we're subtracting the over the overlapping part so if you want to find the union of two probabilities all right the probability of A or B happening you can add the individual probabilities but then you need to subtract the overlapping part anyway we'll see how these formulas get used as we look at specific problems by the way so these are the versions of the formulas where I'm using the words or and the word and okay but everywhere you see the word or I could use a Union symbol all right because it means the same thing and everywhere you see and I could use an intersection symbol so here are the same three formulas written with the Union symbol for or a or B and an intersection symbol but it still means a and B and this still means a or B so please be familiar and comfortable both ways alright anyway those will come up again so let's take a look at the first problem the metal distribution from the 2000 summer Summer Olympic Games is shown in the table below choose one metal wind winner at random okay now problem number one find the probability that the winner won the gold medal given that the winner was from the United States all right so they clearly gave us this word given and when they do that it's um it makes it sort of easy if they're giving us that the winner was from the United States then what you do is you circle that part of the chart alright we're going to circle the united states part of the chart okay for the rest of this problem we can ignore everything else just cross it on out because our universe just got a lot smaller only the united states is going to matter for the rest of the problem now we look back to see what was the probability they were really asking us about okay the problem said find the probability that the winner won the gold medal so all right so gold medal is right here so probability is target over total so our target is gold medal so that's 39 and our total is going to be the sum of these three so 39 plus 25 plus 33 I'm just going to put that in my calculator 39 plus 25 plus 33 that is equal to 97 so that's the total so 39 divided by 97 all right um that probably does not reduce but let's just check it out all right 39 over 97 all right it does not reduce so you could just write the prop whoops you could write the probability just like that of course you could also put it as a decimal if you use a decimal by toggling it you need to write four decimal places point 4:02 one gotta round up and of course we could then write that as a percent very easily forty point twenty one percent probability okay problem number two find the probability that the winner was from the United States given that he or she won a gold medal so let's erase them well that's not the eraser whoops okay find the probability that the winner was from the United States given that he or she won a gold medal so look at what we're given all right because of this given that we're talking about a gold medal then that means let's circle that part of the chart so we're given that he or she won the gold medal all right so we can forget about the rest of the data let's cross it out so we won't be tempted to use it now what are we trying to find again the probability that the winner was from the United States all right target over total our target is United States okay we're being a little egocentric today it's all about us in both problems but that's okay so thirty-nine is a target now out of the total so the total is going to be the total of all of these numbers so I think I'll just do that off camera but I'm going to add up all of these numbers right here all right so the total is 301 so 39 over 301 now embedding this won't reduce either 39 over 301 okay that didn't reduce um as a decimal it would be point one two nine six gotta round up which would be approximately twelve point nine six percent okay so that's how you do the this is called conditional probability when we deal with the given that and we narrow down the universe okay so it looks like we're moving on to a different type of problem all right so this is going to be a nice little review for us a medical research laboratory where a drug that promotes hair growth like you can use that in balding men is being tested the results of the preliminary tests are shown in the table below choose one test subject at random all right now find the probability that a test subjects hair grew given that he used the experimental drug alright are we still doing that given that problems I thought we just mastered that it's kind of hoping we'd have a different type of problem now but okay guess we're doing this I'm given that he used the experimental drug so that means we will circle that part of the chart all right which means we're only looking at this part all right using the using of the drug part a placebo that means a fake pill like a sugar pill that has no effect so anyway we're going to cross that out because that's not what the part that we're given now we look back and say what are we trying to find the probability that a test subjects hair grew okay so target over total hair growth is our target and our total is if we add these together so that's going to be 2,400 if we add those up so probability a target over total that's going to be 1,600 over twenty two thousand four hundred okay we can begin to simplify this mentally alright taking dividing both of these by 100 would be 16 over 24 both of these are divisible by 8 so that would be 2 over 3 so 2/3 so that's the probability right there as a decimal this would be 0.666 7 and as a percent this would be 66.67% okay so there you go for number 3 alright look at number 4 find the probability that a test subject used the placebo given that his hair grew alright look at what we're given alright so we're given that hair grew so let's just focus in on that part of the chart okay given that his hair grew hair growth is this part of the chart we will therefore cross out this part of the chart okay because that's not part of our universe anymore so now what are they asking is for again the probability that the test subject used the placebo the fake drug that doesn't do anything alright so that's our target is placebo so here's the placebo number right there and again so we'll need the total for this row so let's see 28 2800 yeah alright that's the total so target over total placebo over total so 1200 over 2800 okay dividing by a hundred see the two zeros in the numerator and denominator so we can see mentally that this will be 12 out of twenty-eight both of these are divisible by four so if I divide these by 4 4 goes into 12 3 times 4 goes into 28 seven times so that's 3/7 and I could make that to a decimal if I felt like get 3/7 that's point 4 2 8 6 gotta round up all right so that's 42.86 running out of space there okay but it's there trust me 42.86 percent ok that's how we do that all right hopefully we're all done with that type of problem because that's plenty and we're really super good at that type um all right moving on to some of this let's see use the definition of independent events to determine whether the events from rolling a number cube die are independent or dependent okay so the odd event the event odd and the event three or six all right and so in our formula we have events a and B all right and so we have two events here the first event is that we get an odd and the second event is that we either get a three or a six okay so let's call this event a and let's call this event B so we're supposed to prove mathematically whether or not these two events are independent by checking to see if the intersection equals the product of the individual probabilities that's a lot of words but basically we're checking to see if this formula works if it does in pendant if the formula is not true then we'll say dependent okay so maybe underneath where it says independent or dependent we'll right works alright talking about the formula either the formula works for independent or it doesn't work for dependent either it works or not independent dependent anyway so let's figure out the probability of a and B happening okay let's think about this for a second all right so remember event a is rolling odd and event B is rolling a 3 or 6 so our first job is to find the probability of a and B okay as happening at the same time in other words the overlap between these two categories alright you can probably just do this mentally what's the overlap between the odd numbers and the numbers three or six yeah it's just the number three is the only thing that's that these two sets have in common all right odd numbers and the numbers three or six well three is the only one this set only has two numbers in it and three is the odd one so um so three is the only thing that they have in common so basically that what's the probability of rolling a three well there's only the one three so we have one chance in six so one out of six is the probability of a and B happening at the same time because it would have to be a three for that to be true alright and there's only one of those so what we just found was the probability of a and B okay so that's the first part of the formula I replace the intersection sign it means and so I'm just showing that it's the same thing so we found this it turned out to be 1/6 now let's find the probability of a separately okay back to the black screen okay so now we're just finding the probability of a forget about be okay now the probability of a well a is odd so what's the probability of being odd yeah it's one half okay well technically there are three odd numbers on a number Q right one three and five out of a total of six numbers so three and six which you know quickly it reduces down to one half so that's the probability of just a so let's write that down okay so probability of a is 1/2 all right now let's find the probability of just be all right event B is rolling a 3 or 6 so what's the probability of event B well event B has two chances all right there's two ways two targets out of a total of six numbers on the cube so that quickly reduces down to one-third if I divide both of these by two so the probability of B by itself is one-third okay so let's write that down one-third okay so we filled in each of the parts of this formula the probability of a and B the probability of a and the probability of B now we're supposed to check this formula and see if the formula actually works if so then we will declare these events independent so does one-half times one-third equal one-sixth well sure because you just multiply straight across one times one is one two times three is six so the equation does hold um so that means it because it works that means it must be independent these events are independent all right if it doesn't work then it's dependent let's do the same thing for problem number six okay so we have once again we have two events the first event is less and three hmm I think they meant less than three I think that's a typo less than three and event B will be more than five so let's call this event a and we'll call this event event B all right so once again we need to find the intersection you know the overlap between these two sets and then we need to find the individual probabilities and see if they multiply and equal each other so um maybe I won't need to cut over to the black screen this time let's see if we can do it right here so less than three more than five um what are the overlapping numbers between those two sets well there are zero numbers in that in that overlap there's no overlap between these two sets all right there are no numbers that are less than three and at the same time more than five that's not happening so that probability is zero so that the overlap being probability is zero now um so this is not looking good for two being independent now the probability of a probability of being less than three well remember a number Q it goes one two three four five six so as far as being less than three there are two of those numbers 1 & 2 so the probability there is going to be 2 out of 6 and of course 2 out of 6 reduces to 1/3 okay so that first probability is 1/3 I mean I could have left it 2 out of 6 but I just feel like reducing now let's look at probability B by itself more than 5 okay I'm looking at these possibilities again more than 5 there's only one number bigger than 5 that's 6 so the chances of rolling that are 1 out of 6 of course these are not going to multiply together and give us 0 okay 1/3 times 1/6 in fact would be one 18th so it's certainly not zero so that means on this equation is not true so these these events are not independent they are dependent ok because it didn't work all right when it doesn't work it's dependent ok let's do the same thing for number 7 see if I can get that formula cut and pasted my friends ok once again we're going to check out this formula so what do we have the event a will be even and event B will be prime number okay um I'm going to go to the black screen okay so event a was a then and the event B was prime number okay and we are supposed to find the probability right now we're finding we're going to find the probability of a and B but first of all remember we're talking about a six-sided number cube so overall a six-sided number cube goes like this it goes one two three four five or six those are the possibilities okay so um of course the even numbers are two four and six now maybe I should have made those yellow right even numbers are two four and six now the prime numbers prime numbers are the numbers that can only be factored as one and itself now two is a prime number all right it's the only even prime number because two can only be written as one times two it can only be factored as one times two now what what else okay one does not count as a prime number two is a prime number three is a prime number okay four is not a prime number because that can be that's 2 times 2 5 is a prime number and 6 is not a prime number because 2 times 3 so the yellow numbers here these are the even ones the green numbers here these are the Prime's okay so looking back at the formula the first thing we need to find is the probability of a and B in other words the overlap between the two sets okay so what's the overlap between these two sets what do they have in common um this is the only thing they have in common they both have to the number two in common okay so they only have one thing in common so that means that probability is going to be one out of six only one number out of the six total that's what the overlap is one chance in six so let's write that down so the probability of a and B is one-sixth okay now let's check the individual probabilities what's the probability of a okay so now we're for the probability of a we're just looking at these yellows there's three of them alright so the probability for a and I guess I should write this down the probability of a is three chances out of six which is one-half okay great so one-half now how about probability B well probability B has the same thing going on probability of B it also has three chances out of a total of six so that is also one-half okay so the question is are these events independent or dependent yeah these events are dependent all right because 1/2 times 1/2 is 1/4 not 1/6 okay so those events those events are dependent change that equal sign to a not equal sign to emphasize that the formula did not work depending alright um looks like we have one more of these okay um event a is more than 4 and event B is 2 okay so let's find the intersection because once again we are checking this formula to prove whether or not they are dependent or independent so let's start by going to the black screen all right event a is more than 4 okay okay these are the numbers that are more than 4 so the probability actually let me not say the probability of a yet maybe I will anyway the number is more than 4 like we said are 5 & 6 okay for event B I'm sorry what was the event B again I got distracted the event B my friends was simply the number 2 okay so obviously shows al all right this is a strange little situation because what we're supposed to be finding now is the overlap between these two we're supposed to be finding the probability of a and B the overlap 5/6 – um yeah they have nothing in common okay so the probability of picking a number that's both at the same time that's probability of 0 so if we blindly fill that into the formula then on the left side we're going to have to put a 0 the probability of a by itself um we just go back and do my colors colors colors the probability of a by itself well I have two chances all right two chances out of six which equals 1/3 all right the probability of event B by itself well I have one chance for event B so it's one chance in six so I have 1/3 and 1/6 so showing my work properly 1/3 times 1/6 all right clearly zero does not equal 1/3 times 1/6 in fact this would be one eighteenth so what are we going to put right here dependent all right that is it my friends um yeah that's all that's it for this lesson I hope it was helpful and I will see you on 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