Answer to Question #192265 in Statistics and Probability for Milicent

Question

n more than 75sec (5)

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

2.2.1 Exactly 4 have a valid driver’s license? (5) 2.2.2 At least 2 have a valid driver’s license? (5) 2.2.3 More than 9 have a valid driver’s license? 

Answer 1

(2.2)The probability that a student has a valid driver's license is $p=0.3$.

We use binomial distribution for this problem.

Let's assume that a random variable $X$ denotes a number of people that have driver's license.

Now, we have to calculate the following probabilities:

(2.2.1)  Exactly 4 have a valid driver’s license:

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

(2.2.2) At least 2 have a valid driver’s license:

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

$P(X\ge 2)=1-p^{10}-^{10}C_{1}p^9(1-p)^1=1-(0.3)^{10}-10\times0.3^9\times0.7\approx0.9999$

(2.2.3) More than 9 have a valid driver’s license

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