# What is the probability of #X= 6# successes, using the binomial formula?

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Let #n=13, p = .43# .

Let

##### 1 Answer

Jan 7, 2017

#### Explanation:

The binomial formula states that the probability of getting exactly

#P(X=x)=((n),(x))p^x(1-p)^(n-x)," "x=0,1,2,...,n#

where

#((n),(x))# is the number of ways to position#x# successes in a sequence of#n# trials, equal to#(n!)/(x!(n-x)!)# ;#p^x# is the probability of getting those#x# independent successes; and#(1-p)^(n-x)# is the probability of failure for the remaining#n-x# independent trials.

To obtain an answer, we simply plug in the given values of

#P(X=6)=((13),(6))(.43)^6(1-.43)^(13-6)#

#color(white)(P(X=6))=(13!)/(6!(13-6)!)(.43)^6(.57)^7#

#color(white)(P(X=6)) ~~ (1716)(0.006321)(0.015949) #

#color(white)(P(X=6)) ~~ 0.2121 #

So, out of 13 trials, the probability of obtaining exactly 6 successes is