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Chapter 1: Q15SE (page 1)

Suppose that, on a particular day, two persons A and B arrive at a certain store independently of each other. Suppose that A remains in the store for 15 minutes and B remains in the store for 10 minutes. If the time of arrival of each person has the uniform distribution over the hour between 9:00 a.m. and 10:00 a.m., what is the probability that A and B will be in the store at the same time?

Short Answer

Expert verified

probability that A and B will be in the store at the same time is 0.3715

Step by step solution

01

Given information

A remains in the store for 15 minutes

B remains in the store for 10 minutes

02

Calculate the probability that A and B will be in the store at the same time.

If the time of arrival of each person has a uniform distribution over the time between

9 Am and 10 Am.

Let X be the arrival time of 1st person and Y be the arrival time of 2nd person.

X and Y are independent and uniformly distributed over the interval (9 Am, 10 Am)

Then the joint pdf of X and Y is

\(f\left( {x,y} \right) = \left\{ \begin{array}{l}\frac{1}{{3600}}\;\;if\;\;0 < x < 60,\;0 < y < 60\\0\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{array} \right.\)

Lets define \(W = \left| {X - Y} \right|\) which is equal to the probability that the point (X,Y) lies in the shaded region.

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Most popular questions from this chapter

Suppose that four guests check their hats when they arrive at a restaurant, and that these hats are returned to them in a random order when they leave. Determine the probability that no guest will receive the proper hat.

If A, B, and D are three events such that,\({\rm P}\left( {{\rm A} \cup {\rm B} \cup D} \right) = 0.7\)what is the value of\({\rm P}\left( {{{\rm A}^c} \cap {{\rm B}^c} \cap {D^c}} \right)\)?

A physicist makes 25 independent measurements of the specific gravity of a certain body. He knows that the limitations of his equipment are such that the standard deviation of each measurement is 蟽 units.

a. By using the Chebyshev inequality, find a lower bound for the probability that the average of his measurements will differ from the actual specific gravity of the body by less than 蟽/4 units.

b. By using the central limit theorem, find an approximate value for the probability in part (a).

Consider two events A and B with Pr(A) = 0.4 and Pr(B) = 0.7. Determine the maximum and minimum possible values of \(Pr\left( {A \cap B} \right)\) and the conditions under which each of these values is attained.

Consider a state lottery game in which each winning combination and each ticket consists of one set of k numbers chosen from the numbers 1 to n without replacement. We shall compute the probability that the winning combination contains at least one pair of consecutive numbers.

a. Prove that if\({\bf{n < 2k - 1}}\), then every winning combination has at least one pair of consecutive numbers. For the rest of the problem, assume that\({\bf{n}} \le {\bf{2k - 1}}\).

b. Let\({{\bf{i}}_{\bf{1}}}{\bf{ < }}...{\bf{ < }}{{\bf{i}}_{\bf{k}}}\)be an arbitrary possible winning combination arranged in order from smallest to largest. For\({\bf{s = 1,}}...{\bf{,k}}\), let\({{\bf{j}}_{\bf{s}}}{\bf{ = }}{{\bf{i}}_{\bf{s}}}{\bf{ - }}\left( {{\bf{s - 1}}} \right)\). That is,

\(\begin{array}{c}{{\bf{j}}_{\bf{1}}}{\bf{ = }}{{\bf{i}}_{\bf{1}}}\\{{\bf{j}}_{\bf{2}}}{\bf{ = }}{{\bf{i}}_{\bf{2}}}{\bf{ - 1}}\\{\bf{.}}\\{\bf{.}}\\{\bf{.}}\\{{\bf{j}}_{\bf{k}}}{\bf{ = }}{{\bf{i}}_{\bf{k}}}{\bf{ - }}\left( {{\bf{k - 1}}} \right)\end{array}\)

Prove that\(\left( {{{\bf{i}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{i}}_{\bf{k}}}} \right)\)contains at least one pair of consecutive numbers if and only if\(\left( {{{\bf{j}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{j}}_{\bf{k}}}} \right)\)contains repeated numbers.

c. Prove that\({\bf{1}} \le {{\bf{j}}_{\bf{1}}} \le ... \le {{\bf{j}}_{\bf{k}}} \le {\bf{n - k + 1}}\)and that the number of\(\left( {{{\bf{j}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{j}}_{\bf{k}}}} \right)\)sets with no repeats is\(\left( {\begin{array}{*{20}{c}}{{\bf{n - k + 1}}}\\{\bf{k}}\end{array}} \right)\)

d. Find the probability that there is no pair of consecutive numbers in the winning combination.

e. Find the probability of at least one pair of consecutive numbers in the winning combination
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