Although cities encourage carpooling to reduce traffic congestion, most vehicles

Question

Although cities encourage carpooling to reduce traffic congestion, most vehicles carry only one person. For example, 63% of vehicles on the roads are occupied by just the driver. (Round your answers to four decimal places.)

(a) If you choose 10 vehicles at random, what is the probability that more than half (that is, 6 or more) carry just one person?


(b) If you choose 104 vehicles at random, what is the probability that more than half (that is, 53 or more) carry just one person? (Use the normal approximation.)

Explanation / Answer

Binomial 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
a.
P( X < 6) = P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 10 5 ) * 0.63^5 * ( 1- 0.63 ) ^5 + ( 10 4 ) * 0.63^4 * ( 1- 0.63 ) ^6 + ( 10 3 ) * 0.63^3 * ( 1- 0.63 ) ^7 + ( 10 2 ) * 0.63^2 * ( 1- 0.63 ) ^8 + ( 10 1 ) * 0.63^1 * ( 1- 0.63 ) ^9 + ( 10 0 ) * 0.63^0 * ( 1- 0.63 ) ^10   
= 0.2939
P( X > = 6 ) = 1 - P( X < 6) = 0.7061
b.
Normal Approximation to Binomial Distribution
Mean ( np ) =104 * 0.63 = 65.52
Standard Deviation ( npq )= 104*0.63*0.37 = 4.9237
Normal Distribution = Z= X- u / sd                   
P(X > 53) = (53-65.52)/4.9237
= -12.52/4.9237 = -2.5428
= P ( Z >-2.543) From Standard Normal Table
= 0.9945                  

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