ShmoopTube

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finance a la shmoop what is probability distribution this this graph with the

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curvy line line down the middle sitting nicely on an x and y axis it represents [standard distribution bell curve]

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the total sum of probabilities of all these outcomes out here on the far right

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the probability that in three years you sell your screenplay for five million [check changes hands]

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dollars and over here still far the right in the middle is probability you

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sell your screenplay in the next five years for a hundred grand over here just

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right in the middle is the probability you sell your screenplay for a dollar

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but to the shady guy to coffee bean who's promising you and a picture deal

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at Paramount and over here just left in the middle is the probability that

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you're still a barista forever the most likely stuff lives in the middle as we

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slide toward either end things get less and less likely so why is it called a

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distribution well because the potential outcomes ie things like winning a

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lottery selling a screenplay yeah they're kind of the same thing or [happy people with money]

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meeting someone on tinder whose picture was taken less than ten years and twenty [old man walks into park]

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pounds ago carries a range that is the potential outcomes are distributed on a

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long line that then gets visually mapped to explain the character or feelings

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that describes this set of potentialities well the most common

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continuous probability distribution is the normal curve or normal distribution

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you may know it better by its somewhat common nickname this hilly looking thing

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called the bell curve well the mean located in the middle where the peak is [bell curve analysis]

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right there is usually labeled mu which represents a population mean the units

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on each side are plus and minus one two and three standard deviations Sigma is

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the symbol for a population standard deviation well the normal curve was

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developed when researchers started comparing tons of measurements of things

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like heights of giraffes or diameters of plastic lids for drink cups or lengths

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of well just say it got a little competitive there in the lab turns out

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that tons of things both man-made and nature made to end up having a normal

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curve shape to their measurements

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with heights of women they found that a certain height 5 foot 4 inches showed up [woman on a graph]

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more than any other that one height showed up with the greatest probability

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Heights taller than 5'4 and height shorter than 5'4 showed up less often

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well the farther the height was from 5'4 the less likely it was to occur because

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while really tall women in really short women aren't that common and average [short, medium and tall women in a row]

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height women are very common when they plotted the heights and their associated

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probabilities with thousands of results they got a shape that became as the

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normal curve because of the shape of the normal curve 68% of all the possible

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data lands between the first tick marks on each side of the mean plus and minus

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1 Sigma 95 percent of all possible data lands between the second tick marks on

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each side of the mean plus and minus 2 Sigma and ninety-nine point seven

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percent of all the possible data lands between the third tick marks on each

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side of the mean plus and minus the three Sigma there so think about the

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height of women where they would map here and we're going to show you for no

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extra charge where they go on one two and three Sigma there yeah those are the [sleeping man falls out of chair]

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heights well graphically the empirical rule

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shakes out like this in the words of Master Yoda worth memorizing this curve

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is well we can use these percentages to determine how much of the possible data

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plans between different values on the normal curve so let's say we get curious

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and decide to measure the length of every tail of every ring-tailed lemur we [lemurs playing in grass]

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come across which on the streets around here in Silicon Valley is actually more [lab technician measuring tail]

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than you would think all right well then we plot those tail lengths along with

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how often they showed up we'd get a normal curve of tail lengths the mean or

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average tail length would be at the peak in the middle meaning that it was the

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measurement we got most often well the tick marks on the x-axis would be found

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by adding the standard deviation of the tail links to the mean once twice and

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thrice and then subtracting the standard deviation from the mean once twice and

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thrice about 68% of the lemurs we measured would have tail links between

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one sigma negative one sigma there you go

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95% of the lemurs we measured would have tail lengths between two sigma and

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negative two Sigma there we go ninety-nine point seven percent of the

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lemurs we measured would have tail lengths between three sigma there and

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negative three segments right all in that area as another example the machine

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that makes the lids for drink cups doesn't make them the same size every [drinking lid production line]

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time because of variations in the temperature of the plastic and of the

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mold and of the quality of the plastic can't because a butterfly flapped its [butterfly on flower]

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wings in Jamaica the machine will produce lids that are usually around a

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targeted diameter but also slightly large or slightly smaller in fact the

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diameter of plastic lids for a certain size of drink cups are known to be

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normally distributed those lids have a mean diameter of MU equals 3.8 one to

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five inches and a standard deviation of Sigma equals point zero five one inches

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well this means that we can create a normal curve with actual numbers on the

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x axis the mean value in the middle will be the three point eight one to five

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inches will then add point zero five one inches once twice and three times a lady

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to the mean to get the values on the right and subtract point O five one from

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three point eight one two five three times to get the value on the left only

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lids in a range of the sweetspot diameters will fit tightly on the cup

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this sweet spot ranges between three point seven 105 inches and three point

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nine one four five inches well what percentage of lids will be between three

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point seven to 105 inches and three point nine one four five inches in

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diameter and therefore unusable while we're trying to find the percentage of

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lids that will be produced that are between those values at negative two

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sigma 3 point 7 5 and two sigma three point nine one four five yeah well

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according to the empirical rule ninety-five percent of the data lies

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between these two values an empirical rule that's the Empire rule [hand places lid on cup successfully]

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the rule of the trying things out and seeing what happens so that's what the

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data is telling us well 95 percent of the lids produced on

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this machine will be in the sweetspot range and fit tightly on the cups the [surfer holding up]

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empirical rule isn't the only game in town when it comes to normal curving but

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we'll save the other ways to play on the normal curve for a separate video where [monopoly game]

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the normal curve gets the spotlight all to itself well there are other kinds of

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probability distributions that don't cover every possible number decimal

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infraction they're called discrete probability distributions they usually

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hang out in tables and sometimes in formulas turning 18 is great you can

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vote you can be drafted you can buy lottery tickets one quick scratch off

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and you could be on easy street right and maybe not so easy grab a magnifying

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glass peep at the backside of the lottery ticket yep there's a probability

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tribution on the back it shows all the prizes you could win it also shows the

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probabilities or likelihood that you win those prizes and it's a total downer so [woman disappointed with numbers]

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maybe you should ignore it and just scratch and pray you want that new

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jacuzzi with shiatsu massaging jets and it ain't cheap right well specifically [fancy jacuzzi]

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this is a discrete probability distribution which just means we have a

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fixed number of outcomes in this case there are six possible outcomes we can

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win five different dollar amounts and we can also win zilch well check out the

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probability of winning $0 happens 78% of the time you get nothing and then

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there's a one in two thousand or 0.05 percent chance of winning $100 you know

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it would have been better if grandma had just given us the money she used to buy

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the ticket instead of the tickets themselves yeah there are a few other

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kinds of discrete probability distributions here all of them can be

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placed in tables if we want to one specifically has a swaggy formula that

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helps us generate the probabilities for each possible outcome and it's known as

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the binomial probability distribution or BPD for short all to see this thing in

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action we need to have a situation where there are only two things that can

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happen we'll call winning any kind of moolah on [woman happy with being given money]

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that lottery ticket a success we'll call ending up with squat failure well the

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BPD requires exactly two possible outcomes if there are more than two we

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can't use the BPD the BPD also requires that the chance of a success always

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stays the same if there's a twenty two percent chance of winning on the first

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ticket of that kind well there needs to be a twenty two percent chance of [stack of scratch and win tickets]

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winning for all the same kinds of tickets right so it can't be like you're

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picking cards off a deck and all of a sudden there's one less jack so the odds [card dealer fanning cards]

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change every card well the BPD also requires that we don't just spend the

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rest of our lives scratching off those tickets we have to pick a set number of

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tickets we're gonna scratch and you know stick to one meeting those conditions is

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vital if we meet them we can answer questions like if you splurge on ten

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tickets how likely is it that you win on five of them if there's a 22 percent

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chance of winning on each ticket all right well we'd pop in ten for n 5 for K

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and point to 2 for P there all right with all the numbers plugged in we have

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this ugly looking equation yeah grab a calculator there but we need to

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deal with that combinations thing you know the 10 c5 thing yeah it also has [formulas on screen]

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its own formula involving factorials which actually finds all the different

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orders of what could happen on those 10 cards like we could go win win loss loss

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loss win loss win loss win or maybe we get lost lost lost lost win win win win

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lost win yeah well the 10 c5 finds the total

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number of possible ways the card combos could shake out for those of you with a

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TI graphing calculator like a ti-84 or similar we've got you type in the N and [hand using graphing calculator]

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in this case press math go over the PRB menu choose NCR then type in the K it's

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5 in this instance and it should look like 10 NCR 5 on your screen and then

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hit enter well with our combinations number 252 there safely in hand we can

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knock out the rest our answer will tell us how likely it is to win on half of [formulas on screen]

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the 10 tickets you buy when there's a 22 percent chance of winning on each one

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hikes only a 3.75 percent chance of winning on half those tickets you'd have

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been better off spending the money on gas station nachos at least then you'd [woman with nachos]

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have a stomachache to remember your money by maybe just toss your money in

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the trash and then skip the middleman [woman throws nachos away]

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