Question 328703
A drug is known to be 80% effective in curing a certain disease. If four people with the disease are to be given the drug, the probability that more than two will be cured is:
<pre><b>
The binomial probability of exactly x successes in n independent trials, when
the probability of exactly one success in one trial is p, is given by:

{{{(matrix(2,1,n,x))(p)^x*(1-p)^(n-x)}}}

Here there are n trials so n = 4.  The probability of one success (cure) in 1
trial is .8, so p=.8

"More than two" means "three or four", and "or" means to add, so

the probability of 3 or 4 being cured is

{{{(matrix(2,1,4,3))(0.8)^3(1-0.8)^1}}}{{{""+""}}}{{{(matrix(2,1,4,4))(0.8)^4(1-0.8)^0}}}

{{{4!/3!(4-3)!}}}{{{(0.8)^3(.2)}}}{{{""+""}}}{{{4!/4!(4-4)!}}}{{{(0.8)^4(1)}}}

{{{4!/3!1!}}}{{{(.512)(.2)}}}{{{""+""}}}{{{4!/4!0!}}}{{{.4096}}}

{{{4*3*2*1/3*2*1*1}}}{{{.1024}}}{{{""+""}}}{{{4*3*2*1/4*3*2*1*1}}}{{{.4096}}}

{{{4*cross(3)*cross(2)*1/cross(3)*cross(2)*1*1}}}{{{.1024}}}{{{""+""}}}{{{cross(4)*cross(3)*cross(2)*1/cross(4)*cross(3)*cross(2)*1*1}}}{{{.4096}}}

{{{4*.1024}}}{{{""+""}}}{{{.4096}}}

{{{.4086}}}{{{""+""}}}{{{.4086}}}

.8192

You can also find that in the binomial table in your book
with n = 4 and p = .8

Or if you have a TI-83 or TI-84 calculator

On the calculator you have to consider the complement event as
1 - probability that 2 or fewer will be cured:
 
Press ON
Press CLEAR

Type 1-

Press 2ND
Press VARS
Press ALPHA
Press APPS

You should see this -->    1-binomcdf(   
followed by a blinking cursor.
After the parenthesis type this -->  4,.8,2)

You should see this -->    1-binomcdf(4,.8,
                           2)

Press ENTER

You should see the answer .8192

Edwin</pre>