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Chapter 6: Problem 9

Jeanie is a bit forgetful, and if she doesn't make a "to do" list, the probability that she forgets something she is supposed to do is 1 . Tomorrow she intends to run three errands, and she fails to write them on her list. a. What is the probability that Jeanie forgets all three errands? What assumptions did you make to calculate this probability? b. What is the probability that Jeanie remembers at least one of the three errands? c. What is the probability that Jeanie remembers the first errand but not the second or third?

Short Answer

Expert verified
a. The probability that Jeanie forgets all three errands is 1. b. The probability that Jeanie remembers at least one of the three errands is 0. c. The probability that Jeanie remembers the first errand but not the second or third is 0.

Step by step solution

01

Calculate the probability of forgetting all errands

Since the problem states that the probability Jeanie forgets an errand when it is not written down is 1. For three errands, this probability remains the same, i.e., 1.
02

Calculate the probability of remembering at least one errand

The probability of Jeanie remembering at least one errand would be the opposite of her forgetting all errands. Given that the probability of her forgetting all errands is 1, the probability of her remembering at least one errand is \(1 - 1 = 0\).
03

Calculate the probability of Jeanie remembering only the first errand

The question asks for the probability that Jeanie remembers just the first errand, but not the second or third. This scenario, however, is impossible, given the conditions set in the problem. If the errands are not written down, it's stated that Jeanie will forget them all. Thus, the probability of this scenario is 0.

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Key Concepts

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

Probability Theory
Probability theory is fundamentally the mathematics of uncertainty. It's a branch of mathematics dealing with the quantification, analysis, and management of the likelihood and uncertainty of events happening. A probability is a value between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

For instance, if Jeanie has a 100% chance (a probability of 1) of forgetting an errand without a list, the certainty is reflected in the probability value. In her case, for three errands, it seems that the individual probabilities remain constant and therefore Jeanie has a probability of 1 that she will forget each errand. Probability theory would take this information and allow us to predict outcomes, such as her likelihood of forgetting all three errands鈥攅ssential knowledge for planning and decision-making.
Event Probability
The event probability is the measure of the chance of a particular outcome or set of outcomes. It is expressed as a fraction or a decimal ranging from 0 to 1. To calculate the probability of an event, one would typically divide the number of favorable outcomes by the total number of possible outcomes.

For instance, with Jeanie's errands, the event of forgetting an errand is considered to be 'favorable' in this context (although not for Jeanie, perhaps!) and it's given that this event has a probability of 1. So for multiple independent events鈥攆orgetting the first, second, and third errand鈥攖he probability of forgetting all of them (provided they are independent events) would be the product of the probabilities of forgetting each single errand.
Probability Assumptions
Probability assumptions are critical as they underpin our calculations and the conclusions we draw from them. These assumptions might include the independence of events, the total number of possible outcomes, or that all outcomes are equally likely, amongst others.

In our case with Jeanie, a key assumption is that these events are independent; Jeanie's probability of forgetting each errand does not affect her probability of forgetting the others. Importantly, the step-by-step solution of Jeanie's problem presumes that the probability of forgetting is consistent across all errands鈥攁 simplifying assumption. It's also assumed that remembering and forgetting are the only two outcomes for each errand, which is necessary to calculate the complementary probability of remembering at least one errand.

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Most popular questions from this chapter

Many fire stations handle emergency calls for medical assistance as well as calls requesting firefighting equipment. A particular station says that the probability that an incoming call is for medical assistance is .85. This can be expressed as \(P(\) call is for medical assistance \()=.85 .\) a. Give a relative frequency interpretation of the given probability. b. What is the probability that a call is not for medical assistance? c. Assuming that successive calls are independent of one another (i.e., knowing that one call is for medical assistance doesn't influence our assessment of the probability that the next call will be for medical assistance), calculate the probability that both of two successive calls will be for medical assistance. d. Still assuming independence, calculate the probability that for two successive calls, the first is for medical assistance and the second is not for medical assistance. e. Still assuming independence, calculate the probability that exactly one of the next two calls will be for medical assistance. (Hint: There are two different possibilities that you should consider. The one call for medical assistance might be the first call, or it might be the second call.) f. Do you think it is reasonable to assume that the requests made in successive calls are independent? Explain.

\quad\( + Approximately \)30 \%$ of the calls to an airline reservation phone line result in a reservation being made. a. Suppose that an operator handles 10 calls. What is the probability that none of the 10 calls results in a reservation? b. What assumption did you make in order to calculate the probability in Part (a)? c. What is the probability that at least one call results in a reservation being made?

Is ultrasound a reliable method for determining the gender of an unborn baby? Consider the following data on 1000 births, which are consistent with summary values that appeared in the online journal of Statistics Education ("New Approaches to Learning Probability in the First Statistics Course" 120011 ): \begin{tabular}{l|cc} & Ultrasound Predicted Female & Utrasound Predicted Male \\ \hline Actual Gender Is Female & 432 & 48 \\ Actual Gender is Male & 130 & 390 \\ \hline \end{tabular} Do you think that a prediction that a baby is male and a prediction that a baby is female are equally reliable? Explain, using the information in the table to calculate estimates of any probabilities that are relevant to your conclusion.

USA Today (March 15,2001\()\) introduced a measure of racial and ethnic diversity called the Diversity Index. The Diversity Index is supposed to approximate the probability that two randomly selected individuals are racially or ethnically different. The equation used to compute the Diversity Index after the 1990 census was $$ \begin{aligned} 1-&\left[P(W)^{2}+P(B)^{2}+P(A I)^{2}+P(A P I)^{2}\right] \\ & \cdot\left[P(H)^{2}+P(\operatorname{not} H)^{2}\right] \end{aligned} $$ where \(W\) is the outcome that a randomly selected individual is white, \(B\) is the outcome that a randomly selected individual is black, \(A I\) is the outcome that a randomly selected individual is American Indian, \(A P I\) is the outcome that a randomly selected individual is Asian or Pacific Islander, and \(H\) is the outcome that a randomly sclected individual is Hispanic. The explanation of this index stared that 1\. \(\left[P(W)^{2}+P(B)^{2}+P(A I)^{2}+P(A P I)^{2}\right]\) is the probability that two randomly selected individuals are the same race 2\. \(\left[P(H)^{2}+P(\text { not } H)^{2}\right]\) is the probability that two randomly selected individuals are cither both Hispanic or both not Hispanic 3\. The calculation of the Diversity Index treats Hispanic ethnicity as if it were independent of race. a. What additional assumption about race must be made to justify use of the addition rule in the computarion of \(\left[P(W)^{2}+P(B)^{2}+P(A l)^{2}+P(A P I)^{2}\right]\) as the probability that two randomly selected individuals are of the same race? b. Three different probability rules are used in the calculation of the Diversity Index: the Complement Rule, the Addition Rule, and the Multiplication Rule. Describe the way in which each is used.

Many cities regulate the number of taxi licenses, and there is a great deal of competition for both new and existing licenses. Suppose that a city has decided to sell 10 new licenses for \(\$ 25,000\) each. A lottery will be held to determine who gets the licenses, and no one may request more than three licenses. Twenty individuals and taxi companies have entered the lottery. Six of the 20 entries are requests for 3 licenses, nine are requests for 2 licenses, and the rest are requests for a single license. The city will select requests at random, filling as much of the request as possible. For example, the city might fill requests for \(2,3,1,\) and 3 licenses and then select a request for \(3 .\) Because there is only one license left, the last request selected would receive a license, but only one. a. An individual who wishes to be an independent driver has put in a request for a single license. Use simulation to approximate the probability that the request will be granted. Perform at least 20 simulated lotteries (more is better!). b. Do you think that this is a fair way of distributing licenses? Can you propose an alternative procedure for distribution?
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