Road Construction in the Town of Hiawatha Has Presented Some

question 34

Essay

Road construction in the town of Hiawatha has presented some problems for the
traffic engineers. Small businesses such as Bev's Hair Gallery will require an
alternate entrance for their customers. This will in turn create a minor congestion of
traffic nearby because cars will be backed up and the entrance to the parking lot. The
engineers have studied the problem using simulations based on current traffic
patterns. The results from 500 trials are shown below: $$\begin{array} { | c | c | } \hline \boldsymbol { k } & \begin{array} { l } \text { Number of times } \boldsymbol { k } \\\text { cars are backed up } \\\text { at the entrance }\end{array} \\\hline 0 & 250 \\1 & 150 \\2 & 75 \\3 & 25 \\\hline\end{array}$$ Let the random variable k = number of cars backed up at the entrance.
a) Fill in the table below with the estimated probability distribution of k, and sketch
a probability histogram for x. Probability distribution
$\begin{array}{|l|l|}\hline k & \mathrm{P}(x) \\\hline \mathbf{0} & \\\hline \mathbf{1} & \\\hline 2 & \\\hline 3 & \\\hline\end{array}$

Probability histogram

b) Using the estimated probabilities in part (a), calculate the following: i) $P ( k = 1 )$ , the probability that 1 car is backed up at the entrance.
ii) $P ( x < 2 )$ , the probability that fewer than 2 cars are backed up at the entrance
iii) $P ( x \geq 1 )$ , the probability that at least 1 car is backed up at the entrance