Chapter 6 statistics | Revision Server | Page 1

ionic yoke Nov 27, 2022, 5:14 PM

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help me understand it please

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@fierce pumice

fierce pumice Nov 27, 2022, 5:14 PM

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perfect

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so

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ASSUMPTIONS:
a fixed number of "trials", where a trial is just like a run of the event
each trial can only either 'succeed' or 'fail'
the probability of success is constant for each trial
the trial are independent (so no trial impacts the other)

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so just as an example, the dice roll thing from before about getting a 6 or not getting a 6

ionic yoke Nov 27, 2022, 5:15 PM

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yeah I get that

fierce pumice Nov 27, 2022, 5:15 PM

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we can consider getting a 6 to be a 'success'

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as a way to split it up

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ok now

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binomials follow a specific probability distribution

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im gonna write it out then we can talk thru it

ionic yoke Nov 27, 2022, 5:16 PM

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ok

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thanks

fierce pumice Nov 27, 2022, 5:17 PM

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ok so

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yuck

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does the top line look familiar at all?

ionic yoke Nov 27, 2022, 5:18 PM

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yeah this looks familiar but idk what it means

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pretty sure B means binominal

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n is the number of trials

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p is the probability

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I think

fierce pumice Nov 27, 2022, 5:18 PM

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yep perfect

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and the squiggle just means

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"is distributed like"

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so the first line basically reads

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X is distributed like a binomial random variable, with n trials and probability of success p

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ok?

ionic yoke Nov 27, 2022, 5:19 PM

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yeah I understand

fierce pumice Nov 27, 2022, 5:19 PM

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ok awesome

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now if we think logically

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if i have n trials

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the number of successes i can get are gonna be 0, 1, 2, ..., up to n

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as our like possibilities

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yeah?

ionic yoke Nov 27, 2022, 5:20 PM

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yeah

fierce pumice Nov 27, 2022, 5:20 PM

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we cant get like -1 successes, we cant get like n+1 successes , we cant get some weird decimal number of successes

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ok so the second line basically reads

ionic yoke Nov 27, 2022, 5:20 PM

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so it ahs to be an interger

fierce pumice Nov 27, 2022, 5:20 PM

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yep

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between 0 and n

ionic yoke Nov 27, 2022, 5:20 PM

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okk

fierce pumice Nov 27, 2022, 5:20 PM

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so the 2nd line reads as

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the probability we get exactly r successes

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and then that formula

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do you know what nCr means?

ionic yoke Nov 27, 2022, 5:21 PM

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what is r

fierce pumice Nov 27, 2022, 5:21 PM

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r is the number of successes

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were asking lik

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whats the probability we do all n trials and get r successes

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thats what the P(X = r) means

ionic yoke Nov 27, 2022, 5:22 PM

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oh

fierce pumice Nov 27, 2022, 5:22 PM

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its just like a letter to signify that number of successes

ionic yoke Nov 27, 2022, 5:22 PM

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fierce pumice do you know what nCr means?

is it mean number of trials C number of success

fierce pumice Nov 27, 2022, 5:22 PM

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yep

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but do you know what the C means

ionic yoke Nov 27, 2022, 5:23 PM

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no

fierce pumice Nov 27, 2022, 5:23 PM

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its called the choose function which you might have heard of

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ok so

ionic yoke Nov 27, 2022, 5:23 PM

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nope

fierce pumice Nov 27, 2022, 5:23 PM

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itll be on your calculator somewhere but

ionic yoke Nov 27, 2022, 5:23 PM

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yeah it is shift divide

fierce pumice Nov 27, 2022, 5:23 PM

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its basically the number of ways to choose r things out of a list of n things

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so for example if i have say

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ABCDE

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and i wanna choose 2 things

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well i could choose AB, i could choose AE, i could choose BE

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etc

ionic yoke Nov 27, 2022, 5:24 PM

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yeah

fierce pumice Nov 27, 2022, 5:24 PM

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the choose function tells you how many of those options there are

ionic yoke Nov 27, 2022, 5:24 PM

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ok

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so basically

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the total amount of combinations

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e.g.

fierce pumice Nov 27, 2022, 5:24 PM

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perfect exactly

ionic yoke Nov 27, 2022, 5:24 PM

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if I flip a coin twice it could be HH HT TH etc

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okay I get it

fierce pumice Nov 27, 2022, 5:25 PM

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ok so to kinda explain why it shows up in our formula

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if we think about doing these n trials

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with r successes

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if i just call S for success and F for fail

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and lets just do 5 trials and 2 successes again

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just as an example

ionic yoke Nov 27, 2022, 5:25 PM

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ok

fierce pumice Nov 27, 2022, 5:25 PM

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then you can see that these successes could come in lots of different orders like

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SSFFF, or SFFSF, or FFSFS

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etc

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yeah?

ionic yoke Nov 27, 2022, 5:26 PM

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yeah

fierce pumice Nov 27, 2022, 5:26 PM

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so the choose function is here because

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thats all the different ways we can get 2 successes

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we choose 2 successes of the 5 slots we have

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and then we fill in the rest with Fs

ionic yoke Nov 27, 2022, 5:27 PM

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yup

fierce pumice Nov 27, 2022, 5:27 PM

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ok awesome

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and then the probability of succeeding is just p

ionic yoke Nov 27, 2022, 5:27 PM

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fierce pumice and then we fill in the rest with Fs

whats Fs?

fierce pumice Nov 27, 2022, 5:27 PM

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fails

ionic yoke Nov 27, 2022, 5:27 PM

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oh ok

fierce pumice Nov 27, 2022, 5:27 PM

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so we know the probability of success is p just cuz thats what we chose

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and because of the like 'not' rule, the probability of failing is gonna be 1-p

ionic yoke Nov 27, 2022, 5:28 PM

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because P always means probability of success?

fierce pumice Nov 27, 2022, 5:28 PM

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yep

ionic yoke Nov 27, 2022, 5:28 PM

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okay yeah

fierce pumice Nov 27, 2022, 5:28 PM

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fierce pumice

at least thats what i used here

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it might be a number normally

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but getting since we have r successes

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the probability of those bits will be r^p

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and then the other n-r trials fail

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so they have probability (1-p)^(n-r)

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if that makes sense?

ionic yoke Nov 27, 2022, 5:29 PM

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fierce pumice

dont you mean p^r

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not r^p

fierce pumice Nov 27, 2022, 5:29 PM

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i do sorry

ionic yoke Nov 27, 2022, 5:29 PM

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fierce pumice so they have probability (1-p)^(n-r)

yeah I get this

fierce pumice Nov 27, 2022, 5:29 PM

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youre right

ionic yoke Nov 27, 2022, 5:29 PM

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np

fierce pumice Nov 27, 2022, 5:29 PM

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ok awesome so

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thats the like frame work

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so lets maybe take an example

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lets use the dice one again, i roll a dice say 10 times

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and i call a success rolling a 6

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how would you work out the probability of getting 2 6's?

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maybe talk me thru it

ionic yoke Nov 27, 2022, 5:30 PM

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ok

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should I just show you what I would put in my calculator

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and senda photo

fierce pumice Nov 27, 2022, 5:31 PM

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sure

ionic yoke Nov 27, 2022, 5:31 PM

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I could try to talk you through it but let me ensure I get the calculation part right

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one second

fierce pumice Nov 27, 2022, 5:31 PM

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ok sure

ionic yoke Nov 27, 2022, 5:33 PM

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is this right?

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I put it on paint so its clearer maybe

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I will now put inmy calculator and tell you the answer

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is it 0.2907

fierce pumice Nov 27, 2022, 5:33 PM

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no that looks good

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oh idk what the number is but

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that calculation is perfect

ionic yoke Nov 27, 2022, 5:34 PM

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could you double check please?

fierce pumice Nov 27, 2022, 5:34 PM

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0.2907?

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yeah ok we match

ionic yoke Nov 27, 2022, 5:34 PM

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yeah thank god

fierce pumice Nov 27, 2022, 5:34 PM

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so now lets maybe extend our ideas a bit

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how about getting at most 2 6's

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ie 2 or less

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whats ideas do we maybe have for that

ionic yoke Nov 27, 2022, 5:35 PM

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idk I am especially confused for this because my teacher said the sign flips or something

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I dont know sorry

fierce pumice Nov 27, 2022, 5:35 PM

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dont worry about the sign so

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if its 2 or less successes

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what successes are valid?

ionic yoke Nov 27, 2022, 5:36 PM

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0

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1

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2

fierce pumice Nov 27, 2022, 5:36 PM

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perfect so

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why dont you work out the probability for each

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and then since theyre like separate

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you can just add them up

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does that make sense?

ionic yoke Nov 27, 2022, 5:36 PM

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yeah

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that sounds a lot easier than what I was expecting

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one second

fierce pumice Nov 27, 2022, 5:37 PM

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dont bother working out the number but

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what would you calculate

ionic yoke Nov 27, 2022, 5:37 PM

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oh aight

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should I write it out on paint?

fierce pumice Nov 27, 2022, 5:37 PM

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yeah sure

ionic yoke Nov 27, 2022, 5:38 PM

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I would just change the 0 for 1 and then 2

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then add it up

fierce pumice Nov 27, 2022, 5:38 PM

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yep perfect

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so

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heres an exam question

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talk me through what your working would look like

ionic yoke Nov 27, 2022, 5:39 PM

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I would find that 3/5 is 0.6

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so thats the probability of 6 asking for water

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and since there are 10 trials

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because like 10 people

fierce pumice Nov 27, 2022, 5:40 PM

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yep

ionic yoke Nov 27, 2022, 5:41 PM

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then

fierce pumice Nov 27, 2022, 5:41 PM

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so you have 10 trials

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probability for water is 0.6

ionic yoke Nov 27, 2022, 5:41 PM

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is this it for a

fierce pumice Nov 27, 2022, 5:41 PM

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perfect yeah

ionic yoke Nov 27, 2022, 5:41 PM

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lets goo

fierce pumice Nov 27, 2022, 5:41 PM

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so you could simplify that and then calculate it

ionic yoke Nov 27, 2022, 5:42 PM

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yeah

fierce pumice Nov 27, 2022, 5:42 PM

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ok how about the other one?

ionic yoke Nov 27, 2022, 5:42 PM

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ok

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I would do this

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so the probability of 8 asking for water is 0.8

#Chapter 6 statistics