Answer to Question #88471 in Statistics and Probability for tebalelo

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.exactly 4 will have a medical aid
2.at least 2 will have medical aid
3.more than 9 will have a medical aid

Answer 1

The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial.The binomial distribution is used to obtain the probability of observing x successes in n trials, with the probability of success on a single trial denoted by p. The binomial distribution assumes that p is fixed for all trials. The formula for the binomial probability mass function is

$$P(X=x)=\begin{pmatrix} n \\ x \end{pmatrix}p^x(1-p)^{n-x}$$

1. Exactly 4 will have a medical aid.

$$P(X=4)=\begin{pmatrix} 10 \\ 4 \end{pmatrix}0.3^4(1-0.3)^{10-3}=0.200120949$$

2. At least 2 will have a medical aid. 

$$P(X\geq2)=1-P(X<2)=1-P(X=0)-P(X=1)=$$

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

3. More than 9 will have a medical aid. 

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