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Chapter 5: Problem 27

A local downtown arts and crafts shop found from past observation that \(20 \%\) of the people who enter the shop actually buy something. Three potential customers enter the shop. a. How many outcomes are possible for whether the clerk makes a sale to each customer? Construct a tree diagram to show the possible outcomes. (Let \(Y=\) sale \(, \mathbf{N}=\) nosale. \()\) b. Find the probability of at least one sale to the three customers. c. What did your calculations assume in part b? Describe a situation in which that assumption would be unrealistic.

Short Answer

Expert verified
There are 8 outcomes. Probability of at least one sale is 0.488. Assumption: Customers decide independently.

Step by step solution

01

Determine Possible Outcomes

For each customer entering the shop, there are two possible outcomes: a purchase (Y) or no purchase (N). Since there are three customers, we calculate the total number of outcomes as \( 2^3 = 8 \). This is because each customer has two outcomes, and there are three customers. The possible combinations of sales and no-sales can be shown as YYY, YYN, YNY, YNN, NYY, NYN, NNY, NNN.
02

Construct a Tree Diagram

To visualize the possible outcomes, construct a tree diagram. Start with a root node and branch out for the first customer's decision (Y or N). From each of these branches, create two more branches for the second customer's decision (Y or N), and repeat the process for the third customer. You will end up with 8 end branches, each representing one of the possible combinations (e.g. YYY, YYN, etc.).
03

Calculate Probability of Sale

Each customer makes a purchase with a probability of 0.20 (given as 20%). Since decisions are independent, the probability of all customers making no purchase (NNN) is \( (0.80)(0.80)(0.80) = 0.512 \). To find the probability of at least one sale, we use the complement rule: \( 1 - P( ext{NNN}) = 1 - 0.512 = 0.488 \).
04

Assumptions in Probability

The calculation assumes that each customer makes their purchase decision independently of the others. This might not be realistic if customers influence each other's decisions, such as friends shopping together where one friend purchasing might encourage others to do the same.

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Key Concepts

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

Tree Diagram
A tree diagram is a visual tool used to display all possible outcomes of an event. It helps in systematically tracing every possible branch of an event sequence. Imagine a situation like a decision-making process. Here, each decision point generates a new branch.
Visualize yourself in a scenario with three customers entering a shop, where each can either make a purchase (Y) or not (N). This dichotomy creates a tree-like structure:
  • Start from a root, representing the initial state before any customers make a choice.
  • Branch out into two possible outcomes for the first customer: Y or N.
  • Each branch from the first customer extends again into two more branches for the second customer's possible decisions.
  • Finally, each of those branches splits into two final branches for the third customer鈥檚 choices, resulting in a total of 8 combinations (e.g., YYY, YYN, YNY, ...).
Tree diagrams provide not only all possible arrangements but also a powerful way to visually understand complex decision scenarios and outcomes.
Independent Events
Independent events are crucial in probability, especially when determining the likelihood of simultaneous outcomes. Two events are considered independent if the outcome of one event does not influence the outcome of another.
  • In our exercise, each customer has a separate chance of making a purchase (Y) or not (N). Here, every customer's decision is independent of the others.
  • The event where one customer buys or does not buy something does not alter probabilities for the other customers.
  • Thus, the probability of a specific sequence of decisions can be calculated by multiplying their individual probabilities. For instance, the chance of all customers not buying (NNN) is computed as: \( (0.80) \times (0.80) \times (0.80) = 0.512 \).
Understanding independent events helps us confidently apply probability rules, secure that one decision doesn鈥檛 sway another.
Complement Rule
The complement rule in probability is a technique used to find the probability of getting at least one occurrence of an event. It鈥檚 especially useful when directly calculating the event is complex.
  • The complement of an event is simply the opposite of what you are trying to find. In our case, the complement to finding at least one sale is all three customers making no purchase 鈥� calculated as \( (0.80) \times (0.80) \times (0.80) = 0.512 \).
  • So, instead of finding probabilities for numerous individual potential buy scenarios, you calculate the probability of no buys and subtract it from 1. This gives: \( 1 - 0.512 = 0.488 \).
  • This approach is efficient and simplifies the process, particularly for larger and complex scenarios.
Using the complement rule reduces effort, ensuring a straightforward path to arrive at the probability of extensive possibilities.

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Most popular questions from this chapter

Labor force In \(2014,\) a sample of 1925 Americans revealed that about \(20.5 \%\) of them belong to the government sector. \(7.5 \%\) of these are part-time employees, \(60 \%\) are full-time employees, and \(32.5 \%\) are retired. a. Define events and identify which of these four probabilities refer to conditional probabilities. b. Find the probability that an American adult in this sample is a full-time government employee.

Your teacher gives a true-false pop quiz with 10 questions. a. Show that the number of possible outcomes for the sample space of possible sequences of 10 answers is \(1024 .\) b. What is the complement of the event of getting at least one of the questions wrong? c. With random guessing, show that the probability of getting at least one question wrong is approximately \(0.999 .\)

The digits in \(9 / 11\) add up to \(11(9+1+1)\), American Airlines flight 11 was the first to hit the World Trade Towers (which took the form of the number 11 ), there were 92 people on board \((9+2=11),\) September 11 is the 254 th day of the year \((2+5+4=11),\) and there are 11 letters in Afghanistan, New York City, the Pentagon, and George W. Bush (see article by L. Belkin, New York Times, August 11,2002 ). How could you explain to someone who has not studied probability that, because of the way we look for patterns out of the huge number of things that happen, this is not necessarily an amazing coincidence?

In criminal trials (e.g., murder, robbery, driving while impaired, etc.) in the United States, it must be proven that a defendant is guilty beyond a reasonable doubt. This can be thought of as a very strong unwillingness to convict defendants who are actually innocent. In civil trials (e.g., breach of contract, divorce hearings for alimony, etc.), it must only be proven by a preponderance of the evidence that a defendant is guilty. This makes it easier to prove a defendant guilty in a civil case than in a murder case. In a high- profile pair of cases in the mid 1990 s, O. J. Simpson was found to be not guilty of murder in a criminal case against him. Shortly thereafter, however, he was found guilty in a civil case and ordered to pay damages to the families of the victims. a. In a criminal trial by jury, suppose the probability the defendant is convicted, given guilt, is \(0.95,\) and the probability the defendant is acquitled, given innocence, is 0.95 . Suppose that \(90 \%\) of all defendants truly are guilty. Given that a defendant is convicted, find the probability he or she was actually innocent. Draw a tree diagram or construct a contingency table to help you answer. b. Repeat part a, but under the assumption that \(50 \%\) of all defendants truly are guilty. c. In a civil trial, suppose the probability the defendant is convicted, given guilt is 0.99 , and the probability the defendant is acquitted, given innocence, is \(0.75 .\) Suppose that \(90 \%\) of all defendants truly are guilty. Given that a defendant is convicted, find the probability he or she was actually innocent. Draw a tree diagram or construct a contingency table to help you answer.

Before the first human heart transplant, Dr. Christiaan Barnard of South Africa was asked to assess the probability that the operation would be successful. Did he need to rely on the relative frequency definition or the subjective definition of probability? Explain.
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