Question

A regional bus between Fort Collins and Denver mostly uses the high-occupancy vehicle (HOV) lane but it does have to cross two intersections. Let X be the number of stops the bus must make and the PMF is given below.
X 0 1 2
p(x) 0.24 0.15 0.61

Round all solutions to three decimal places.
Now suppose the bus makes a round trip, so the sample size is 2. We are interested in Z=X1 + X2, where X1 is the number of stops on the trip from Fort Collins to Denver of the trip and X2 is the number of stops on the trip from Denver to Fort Collins. Note that Z can take on values 0, 1, 2, 3, 4.

Compute P(Z=0), that is, compute the probability that no stops are made in the round trip (Z=X1 + X2=0).

Answer 1

Answer:

0.058 is the required probability.

Step-by-step explanation:

We are given the following PMF in the question:

   x:       0        1          2

p(x):    0.24    0.15    0.61

$Z=X_1 + X_2$

Z can take values 0, 1, 2, 3, 4.

We have to compute the value of P(Z=0)

$(Z=X_1+X_2) \\P(Z=0)=P(X1=0)+P(X2=0) \\P(Z=0)=0.24\times 0.24\\P(Z=0) =0.0576\approx 0.058$

Thus,

$P(Z = 0) = 0.058$