Question #179447

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?


1
Expert's answer
2021-04-15T06:49:57-0400

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

  1. P(X=4)=C104p4(1p)6=10!4!6!(0.3)4(0.7)60.2001P(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(X2)=1P(X<2)=1P(X=0)P(X=1)=P(X\geq2)=1-P(X<2)=1-P(X=0)-P(X=1)=

=1p10C101p9(1p)=1(0.3)10100.390.70.9999=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)=(1p)10=(0.7)100.0282P(X>9)=P(X=10)=(1-p)^{10}=(0.7)^{10}\approx0.0282


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