[tex]\binom{18}{0}(0.7)^{18}+\binom{18}{1}(0.3)^1(0.7)^{17}[/tex]
This will be a binomial probability, since there are two options - either cured or not cured.
We want the probability that less than or equal to one person is cured. This means we want the probability that either 0 people are cured or 1 person is cured.
To find the probability of 0 people being cured, we take a combination of 18 people chosen 0 at a time:
[tex]\binom{18}{0}[/tex]
We multiply this by the probability of someone being cured, 0.3, raised to the number of people being cured, 0:
[tex]\binom{18}{0}(0.3)^0\\\\=\binom{18}{0}(1)=\binom{18}{0}[/tex]
Lastly we multiply this by the probability of someone not being cured, 1-0.3, raised to the number of people not being cured, 18:
[tex]\binom{18}{0}(1-0.3)^{18}\\\\=\binom{18}{0}(0.7)^{18}[/tex]
To find the probability of 1 person being cured, we take a combination of 18 people chosen 1 at a time:
[tex]\binom{18}{1}[/tex]
We multiply this by the probability of someone being cured, 0.3, raised to the number of people being cured, 1:
[tex]\binom{18}{0}(0.3)^1[/tex]
Lastly we multiply this by the probability of someone not being cured, 1-0.3, raised to the number of people not being cured, 17:
[tex]\binom{18}{0}(0.3)^1(1-0.3)^{17}\\\\=\binom{18}{0}(0.3)^1(0.7)^{17}[/tex]
