p = .8 = probability that a person who gets the disease will recover.
q = .2 = probability that a person who gets the disease will not recover.
q = 1 - p
p(x out of n will recover) = nCx * p^x * q^(n-x)
n = 7
x = 3
nCx = combination formula of n! / (x! * (n-x)!)
formula becomes:
7C3 * .8^3 * .2^4 = 35 * .512 * .0016 = .028672
this is the probability that exactly 3 will recover.
the formula used is the binomial probability formula.
the probability of exactly 0,1,2,3,4,5,6,7 recovering is shown below:
n = 7
p = 0.8
q = 0.2
x p^x q^(n-x) nCx p(x)
0 1 0.0000128 1 0.0000128
1 0.8 0.000064 7 0.0003584
2 0.64 0.00032 21 0.0043008
3 0.512 0.0016 35 0.028672 *****
4 0.4096 0.008 35 0.114688
5 0.32768 0.04 21 0.2752512
6 0.262144 0.2 7 0.3670016
7 0.2097152 1 1 0.2097152
total probability = 1