Question 243421
ok. if i have an eighty percent chance of selecting the right outcome for a particular event, then i know the probability for selecting the right outcome for four, five or six events in a row. but what is the probability of selecting the right outcome four out of six times, or five out of six times. I'm getting a little confused when the outcomes are not in a row. Because four out of six could be four hits in a row or it could be miss,hit,hit,hit,miss,hit and so on. please help. thanks, tim
<pre><font size = 4 color = "indigo"><b>
You are talking about binomial probabilities.

It doesn't matter whether the right outcomes are in a row or not.

If the probability of 1 success in 1 trial is p, then the probability
of exactly x successes in n trials (regardless of the order in which
the successes occur) is given by this formula:

{{{"C(n,x)"*p^x*(1-p)^(n-x)}}}

Sometimes {{{"C(n,x)"}}} is written {{{nCx}}} and sometimes 
it's written {{{(matrix(2,1,n,x))}}}.  Sometimes it is called
"n choose x" and sometimes "the combinations of n things
taken x at a time", and sometimes "the n,x binomial coefficient".
But however it's written and whatever
it's called it means {{{  n!/(x!(n-x)!) }}}  or

{{{(n/x)((n-1)/(x-1))((n-2)/(x-2))}}}{{{"..."}}}{{{((n-x+1)/1)}}}

In your problem, the probability of getting exactly 4 successes 
out of 6 trials, when the probability of exactly 1 success in 1 
trial is .8 can be gotten this way:

n = 6, p = .8, x = 4 

{{{(6/4)(5/3)(4/2)(3/1)(.8)^4(1-.8)^2= .24576}}}

In your other problem, the probability of getting exactly 5 successes 
out of 6 trials, when the probability of exactly 1 success in 1 
trial is .8 can be gotten this way:

n=6, p = .5, x = 5

{{{(6/5)(5/4)(4/3)(3/2)(2/1)(.8)^5(1-.8)^1= .393216}}}

You can also use a TI-84 calculator to find this

Press CLEAR
Press 2nd
Press VARS
Press ALPHA
Press MATH

You will see this on the screen:   

binompdf(

type this  6,.8,4) after it and you will have this on the screen:

binompdf(6,.8,4)

Press ENTER

Read .24576

---------------

For the other one replace the 4 with a 5, and
you'll read .393216

Edwin</pre>