Finance: What is probability distribution?

What is probability distribution? Statistical analysis is a study that compiles and breaks down data gathered from a wide range of categories and criteria. When attempting to assess the impact of a variable on all of the different possible outcomes as the variable relates to a host of other factors and scenarios, there are certain outcomes that are more likely than others. Probability distribution is the process of organizing these outcomes in a graph or other display to show these outcomes’ likelihood when referencing the constant factors vs. the variable.

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Transcript

00:26

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

00:30

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

00:33

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

00:37

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

00:41

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

00:45

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

00:50

distribution well because the potential outcomes ie things like winning a

00:54

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

00:57

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

01:01

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

01:06

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

01:11

that describes this set of potentialities well the most common

01:14

continuous probability distribution is the normal curve or normal distribution

01:20

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

01:24

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

01:28

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

01:33

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

01:39

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

01:43

developed when researchers started comparing tons of measurements of things

01:47

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

01:51

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

01:55

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

01:59

curve shape to their measurements

02:04

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

02:08

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

02:12

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

02:17

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

02:22

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

02:26

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

02:30

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

02:34

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

02:39

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

02:43

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

02:49

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

02:53

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

02:58

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

03:02

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

03:05

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

03:09

heights well graphically the empirical rule

03:11

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

03:17

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

03:21

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

03:25

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

03:29

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

03:33

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

03:36

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

03:40

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

03:43

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

03:46

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

03:50

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

03:54

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

03:58

one sigma negative one sigma there you go

04:01

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

04:05

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

04:09

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

04:13

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

04:16

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

04:20

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

04:23

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

04:26

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

04:30

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

04:34

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

04:37

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

04:42

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

04:48

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

04:51

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

04:55

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

04:59

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

05:03

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

05:07

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

05:12

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

05:15

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

05:19

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

05:22

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

05:26

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

05:30

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

05:34

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

05:38

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

05:42

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

05:45

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

05:48

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

05:52

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

05:55

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

05:59

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

06:03

probability distributions that don't cover every possible number decimal

06:06

infraction they're called discrete probability distributions they usually

06:10

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

06:14

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

06:19

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

06:23

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

06:27

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

06:30

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

06:35

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

06:38

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

06:42

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

06:46

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

06:51

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

06:55

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

07:00

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

07:06

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

07:09

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

07:12

kinds of discrete probability distributions here all of them can be

07:15

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

07:19

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

07:23

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

07:29

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

07:32

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

07:34

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

07:39

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

07:44

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

07:49

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

07:53

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

07:56

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

08:00

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

08:03

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

08:07

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

08:10

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

08:14

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

08:19

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

08:24

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

08:29

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

08:33

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

08:37

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

08:42

its own formula involving factorials which actually finds all the different

08:47

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

08:52

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

08:58

lost win yeah well the 10 c5 finds the total

09:01

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

09:05

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

09:11

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

09:18

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

09:23

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

09:27

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

09:32

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

09:36

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

09:41

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

09:45

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

09:48

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