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Chapter 13: Problem 35

A survey found that \(73 \%\) of Americans have a home phone, \(83 \%\) have a cell phone and \(58 \%\) of people have both. a. If a person has a home phone, what's the probability that they have a cell phone also? b. Are having a home phone and a cell phone independent events? Explain. c. Are having a home phone and a cell phone mutually exclusive? Explain.

Short Answer

Expert verified
a. The probability that a person has a cell phone given they have a home phone is approximately \(79.4\%\). b. Having a home phone and a cell phone are not independent events. c. Having a home phone and a cell phone are not mutually exclusive events.

Step by step solution

01

Define the probabilities

Let's define the following probabilities: Let P(H) be the probability a person has a home phone, which is \(73\%\). Let P(C) be the probability a person has a cell phone, which is \(83\%\). And P(H 鈭� C) be the probability a person has both a home phone and a cell phone, which is \(58\%\).
02

Calculate the conditional probability

The conditional probability that they have a cell phone given they have a home phone is P(C|H), which is the probability of the intersection of C and H, divided by the probability of H. So, P(C|H) = P(C 鈭� H) / P(H) = \(58\% / 73\% = 0.794\).
03

Check for independence of events

Two events A and B are independent if and only if P(A 鈭� B) = P(A)P(B). So, let's calculate P(H)P(C) = \(73\% * 83\% = 0.606\). As we see, P(H)P(C) is not equal to P(H 鈭� C). Thus, the two events are not independent.
04

Check for mutually exclusiveness

Two events A and B are mutually exclusive if and only if P(A 鈭� B) = 0. Here, P(H 鈭� C) = \(58\%\), which is not equal to 0. Thus, the events are not mutually exclusive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability
Conditional probability is the likelihood of an event occurring given that another event has already happened. In this context, we determine the probability of an American having a cell phone given they own a home phone.

To find this, we use the formula:
  • The probability of the second event occurring (cell phone user), given the first event (home phone user).
  • It's represented as \( P(C|H) \).
This formula is derived from dividing the probability of both events (intersection) by the probability of the first event.

Here, \( P(C|H) = \frac{P(C \cap H)}{P(H)} \). For this exercise, it calculates to \( \frac{58\%}{73\%} = 0.794 \). This means there's a 79.4% chance that someone with a home phone also has a cell phone.
Independence of Events
Two events are considered independent when the occurrence of one event does not affect the occurrence of the other.

In mathematical terms, events A and B are independent if \( P(A \cap B) = P(A) \times P(B) \). This indicates no influence between the two events.
  • For example, flipping a coin twice where the result of the first flip doesn't affect the second.
For the problem with home phones and cell phones, we calculate:
\[P(H \cap C) = 0.58 \]Comparing with \( P(H) \times P(C) = 0.73 \times 0.83 = 0.606 \), which is not equal.

Thus, because \( P(H \cap C) eq P(H) \times P(C) \), the events are not independent. The probability of owning a cell phone is influenced by owning a home phone.
Mutually Exclusive Events
Mutually exclusive events cannot happen at the same time. If one event occurs, the other cannot. In terms of probabilities, two events A and B are mutually exclusive if \( P(A \cap B) = 0 \).

This is often illustrated with situations like tossing a coin where the result can either be heads or tails 鈥� not both.
  • In your daily routine, either you're awake or asleep, not both.
In our home phone and cell phone scenario, the intersection probability \( P(H \cap C) \) is given as 0.58, not zero.

Therefore, these events are not mutually exclusive, as it's entirely possible for someone to have both phones at the same time.
Intersection of Events
The intersection of events, denoted as \( A \cap B \), represents the scenario where both events occur simultaneously. It's the overlap of two separate events.

For instance, if one event is owning a home phone and the other is owning a cell phone, the intersection is the probability of someone owning both.
  • Think of it as the common section in a Venn diagram where both circles overlap.
Calculating the intersection helps in understanding scenarios where both conditions are met.

In this survey case, \( P(H \cap C) = 58\% \). It's a substantial overlap, showing most people who have a home phone also carry a cell phone, illustrating the compatibility and relevance of the events in study.

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Most popular questions from this chapter

Suppose that the information in Exercise 9 had been presented in the following way. Facebook reports that \(70 \%\) of its users are from outside the United States. Of the U.S. users, two-thirds log on every day. Of the non-U.S. users, three-sevenths log on every day. Draw a tree for this situation. Why is the tree better than the table in this case? Where are the joint probabilities found?

An airline offers discounted "advance-purchase" fares to customers who buy tickets more than 30 days before travel and charges "regular" fares for tickets purchased during those last 30 days. The company has noticed that \(60 \%\) of its customers take advantage of the advance-purchase fares. The "no- show" rate among people who paid regular fares is \(30 \%,\) but only \(5 \%\) of customers with advance-purchase tickets are no-shows. a. What percent of all ticket holders are no-shows? b. What's the probability that a customer who didn't show had an advance- purchase ticket? c. Is being a no-show independent of the type of ticket a passenger holds? Explain.

PewInternet reported in January of 2014 that \(32 \%\) of U.S. adults own at least one e-reader, and that \(28 \%\) of U.S. adults read at least one e-book in the previous year (and thus, presumably, owned an ereader). Overall, \(76 \%\) of U.S. adults read at least one book (electronic or otherwise) in the previous year. a. Explain how these statistics indicate that owning an e-reader and reading at least one book in the previous year are not independent. b. What's the probability that a randomly selected U.S. adult has an ereader but didn't use it to read an e-book in the previous year?

You pick three cards at random from a deck. Find the probability of each event described below. a. You get no aces. b. You get all hearts. c. The third card is your first red card. d. You have at least one diamond.

A university requires its biology majors to take a course called BioResearch. The prerequisite for this course is that students must have taken either a statistics course or a computer course. By the time they are juniors, \(52 \%\) of the biology majors have taken statistics, \(23 \%\) have had a computer course, and \(7 \%\) have done both. a. What percent of the junior biology majors are ineligible for BioResearch? b. What's the probability that a junior biology major who has taken statistics has also taken a computer course? c. Are taking these two courses disjoint events? Explain. d. Are taking these two courses independent events? Explain.
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