Answer to Question #88207 in Statistics and Probability for Marine Smit

Question

It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers:
1.1 At least 2 will have a medical aid
1.2. More than 9 will have a medical aid

Answer 1

The binomial probability formula is the result of getting exactly k successes in n number of trials.

$$P(k \ success\ in\ n \ trials)=\begin{pmatrix} n \\ k \end{pmatrix}p^k(1-p)^{n-k}$$

We have that

$$n=10, p=0.3$$

1.1 At least 2 will have a medical aid 

$$P(X\geq 2)=1-(P(X=0)+P(X=1))=$$

$$=1-\begin{pmatrix} 10 \\ 0 \end{pmatrix}0.3^0(1-0.3)^{10-0}-\begin{pmatrix} 10 \\ 1 \end{pmatrix}0.3^1(1-0.3)^{10-1}\approx 0.850692$$

1.2. More than 9 will have a medical aid

$$P(X> 9)=P(X=10)=\begin{pmatrix} 10 \\ 10 \end{pmatrix}0.3^{10}(1-0.3)^{10-10}\approx 0.000006$$