Answer to Question #179447 in Statistics and Probability for Mathews

Question

If it is estimated that 30% of all students who fall under the Business Economics and management departmebt at Richfield have driver's license. What is the probability of this in a sample of 10 students?

1. Exactly 4 have a driver's license?

2. At least 2 have a valid driver's license?

3. More than 9 have a valid driver's license?

Answer 1

The probability that a student has a driver's license is $p=0.3$. We will use a binomial distribution for this problem. Assume that a random variable $X$ denotes a number of people that have driver's license. We have to calculate the following probabilities:

1. $P(X=4)=C_{10}^4p^4(1-p)^6=\frac{10!}{4!6!}(0.3)^4(0.7)^6\approx0.2001$ (it is rounded to 4 decimal places)
2. $P(X\geq2)=1-P(X<2)=1-P(X=0)-P(X=1)=$

$=1-p^{10}-C_{10}^1p^9(1-p)=1-(0.3)^{10}-10\cdot0.3^9\cdot0.7\approx0.9999$

3. $P(X>9)=P(X=10)=(1-p)^{10}=(0.7)^{10}\approx0.0282$