Question

In the State of Texas, pickup trucks account for 23% of the State's registered vehicles. If 100 registered vehicles are selected at random, answer the following 5 questions:

1) What is the probability that exactly 10 of the selected vehicles are pickup trucks?

2) What is the probability that 20 or less of the selected vehicles are pickup trucks?

3) What is the probability that 30 or more of the selected vehicles are pickup trucks?

4) What is the expected number of pickup trucks?

5) If 1000 vehicles are selected at random, what is the expected number of pickup trucks?

Answer 1

BIONOMIAL DISTRIBUTION
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is executed
p = success probability
mean = 100 * 0.23
= 23
II.
variance = npq
where
n = total number of repetitions experiment is executed
p = success probability
q = failure probability
variance = 100 * 0.23 * 0.77
= 17.71
III.
standard deviation = sqrt( variance ) = sqrt(17.71)
=4.2083

1.
the probability that exactly 10 of the selected vehicles are pickup trucks
P( X = 10 ) = ( 100 10 ) * ( 0.23^10) * ( 1 - 0.23 )^90
= 0.0004

2.
the probability that 20 or less of the selected vehicles are pickup trucks
P( X < = 20) = P(X=20) + P(X=19) + P(X=18) + P(X=17) + P(X=16) + P(X=15) + P(X=14) + P(X=13) + P(X=12) + P(X=11) + P(X=10)
= ( 100 20 ) * 0.23^20 * ( 1- 0.23 ) ^80 + ( 100 19 ) * 0.23^19 * ( 1- 0.23 ) ^81 + ( 100 18 ) * 0.23^18 * ( 1- 0.23 ) ^82 + ( 100 17 ) * 0.23^17 * ( 1- 0.23 ) ^83 + ( 100 16 ) * 0.23^16 * ( 1- 0.23 ) ^84 + ( 100 15 ) * 0.23^15 * ( 1- 0.23 ) ^85 + ( 100 14 ) * 0.23^14 * ( 1- 0.23 ) ^86 + ( 100 13 ) * 0.23^13 * ( 1- 0.23 ) ^87 + ( 100 12 ) * 0.23^12 * ( 1- 0.23 ) ^88 + ( 100 11 ) * 0.23^11 * ( 1- 0.23 ) ^89 + ( 100 10 ) * 0.23^10 * ( 1- 0.23 ) ^90
= 0.2811

3.
the probability that 30 or more of the selected vehicles are pickup trucks
P( X < 30) = P(X=29) + P(X=28) + P(X=27) + P(X=26) + P(X=25) + P(X=24) + P(X=23) + P(X=22) + P(X=21) + P(X=20)   
= ( 100 29 ) * 0.23^29 * ( 1- 0.23 ) ^71 + ( 100 28 ) * 0.23^28 * ( 1- 0.23 ) ^72 + ( 100 27 ) * 0.23^27 * ( 1- 0.23 ) ^73 + ( 100 26 ) * 0.23^26 * ( 1- 0.23 ) ^74 + ( 100 25 ) * 0.23^25 * ( 1- 0.23 ) ^75 + ( 100 24 ) * 0.23^24 * ( 1- 0.23 ) ^76 + ( 100 23 ) * 0.23^23 * ( 1- 0.23 ) ^77 + ( 100 22 ) * 0.23^22 * ( 1- 0.23 ) ^78 + ( 100 21 ) * 0.23^21 * ( 1- 0.23 ) ^79 + ( 100 20 ) * 0.23^20 * ( 1- 0.23 ) ^80
= 0.9357
P( X > = 30 ) = 1 - P( X < 30) = 0.0643

4.
the expected number of pickup trucks
mean = 100 * 0.23
= 23

5.
If 1000 vehicles are selected at random, what is the expected number of pickup trucks
mean = 1000 * 0.23
= 230