# 20.2 A certain market has both an express checkout line and a superexpress checkout line. Let X1...

## Question:

A certain market has both an express checkout line and a superexpress checkout line. Let X1 denote the number of customers in the express line at a particular time of day and let X2 denote the number of customers in the superexpress line at the same time of day. Suppose the joint pdf of X1 and X2 is described in the following table

X1

0 | 1 | 2 | 3 | ||

0 | 0.05 | 0.20 | 0 | 0 | |

1 | 0.15 | 0.10 | 0.03 | 0.02 | |

X2 | 2 | 0.05 | 0.07 | 0.10 | 0.03 |

3 | 0 | 0.03 | 0.04 | 0.03 | |

4 | 0 | 0 | 0.03 | 0.07 |

a.Find the chance there is exactly one customer in each line: P(X1 = 1, X2 = 1).

b.What is the probability that in the superexpress lane is three less than double the number of people in the express lane or P(X2 = 2X1 - 3)?

## Tables Are Key:

Tables that list all possible combinations and their probabilities make questions involving probability much easier. If you are ever stuck on a probability question and do not know how to approach it, a safe starting choice is to list all possible combinations and their probabilities.

## Answer and Explanation:

For part **(a)**, we can directly use the table to find this. Thus,

{eq}P(X1 = 1, X2 = 1) = 0.10 {/eq}

For part **(b)**, note that for X2 to have a possible value, you need to have X1 equal either 2 or 3. You can see that if X1 equals 0 or 1, the value for X2 will be negative and not possible, thus having a probability of 0. Since {eq}2(2) - 3 = 1 {/eq} and {eq}2(3) - 3 = 3 {/eq}, we can conclude that {eq}P(X2 = 2X1 - 3) = P(X1 = 2, X2 = 1 \, \text{or} \, X1 = 3, X2 = 3) {/eq}.

Since these events are mutually exclusive, we can add them to find the probability of either of them occurring. Therefore,

{eq}P(X1 = 2, X2 = 1 \, \text{or} \, X1 = 3, X2 = 3) = P(X1 = 2, X2 = 1) + P(X1 = 3, X2 = 3) = 0.03 + 0.03 = 0.06 {/eq}

and so we find that {eq}P(X2 = 2X1 - 3) = 0.06 {/eq}.

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from 6th-8th Grade Math: Practice & Review

Chapter 48 / Lesson 5