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Chapter 4: Problem 46

Answer questions true or false and give a brief explanation for each answer. Hint: Review the summary of basic probability rules.If \(A\) and \(B\) are mutually exclusive, they must also be independent.

Short Answer

Expert verified
False: Mutually exclusive events are not independent unless one or both occur with zero probability.

Step by step solution

01

Understanding Mutually Exclusive Events

Two events, \(A\) and \(B\), are said to be mutually exclusive if they cannot happen at the same time. This means that if one event occurs, the other cannot. Mathematically, if \(A\) and \(B\) are mutually exclusive, then \(P(A \cap B) = 0\).
02

Understanding Independent Events

Two events, \(A\) and \(B\), are independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, if \(A\) and \(B\) are independent, then \(P(A \cap B) = P(A) \times P(B)\).
03

Analyzing the Relationship

If \(A\) and \(B\) are mutually exclusive, then \(P(A \cap B) = 0\). For them to be independent, \(P(A \cap B) = P(A) \times P(B)\) should be true, but since mutually exclusive events imply \(P(A \cap B) = 0\), and unless one or both events have probability zero, \(P(A) \times P(B)\) is not zero generally. Thus, mutually exclusive events cannot be independent unless one or both have a zero probability.

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Key Concepts

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

Mutually Exclusive Events
In probability, mutually exclusive events are those which cannot happen at the same time. Imagine you have a single six-sided die, and you are asked to roll either a 2 or a 3. Both results cannot occur simultaneously, making them mutually exclusive. In mathematical terms, this is expressed as \(P(A \cap B) = 0\), meaning there is no overlap in occurrence.

  • For example, when flipping a coin, getting a "heads" and "tails" simultaneously is impossible.
  • Thus, the probability of both events occurring together is zero.
This concept is essential in understanding why mutually exclusive events differ significantly from another category known as independent events.
Independent Events
Independent events are events where the occurrence of one does not affect the probability of the other. Think of drawing a card from a deck, replacing it, and then drawing again. The probability of drawing an Ace for both events remains unchanged regardless of previous outcomes. This mathematical relationship is shown as \(P(A \cap B) = P(A) \times P(B)\).

  • This means whether event A has occurred provides no information about the likelihood of event B.
  • For instance, rolling a die does not influence the result of flipping a coin; these are independent events.
It is crucial to note that independent events can both occur, or neither occur, or one can occur while the other does not, all without affecting each other's probabilities.
Probability Rules
Probability rules provide the foundation for analyzing how likely events are to happen. For mutually exclusive events, the addition rule is vital: the probability of either event A or B occurring is the sum of their individual probabilities, \(P(A \cup B) = P(A) + P(B)\), because they cannot happen simultaneously.

  • For independent events, the multiplication rule comes into play, as both can happen together: \(P(A \cap B) = P(A) \times P(B)\).
  • Understanding the zero probability of mutually exclusive events co-occurring and the product of probabilities in independent events helps in identifying how these principles guide event analysis.
These rules allow you to calculate complex probabilities and understand event relationships better, leading to more informed predictions.

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Most popular questions from this chapter

If two events are mutually exclusive, can they occur concurrently? Explain.

You draw two cards from a standard deck of 52 cards without replacing the first one before drawing the second. (a) Are the outcomes on the two cards independent? Why? (b) Find \(P(3\) on 1 st card and 10 on 2 nd ). (c) Find \(P(10\) on 1 st card and 3 on 2 nd ). (d) Find the probability of drawing a 10 and a 3 in either order.

Alcobol Recovery The Eastmore Program is a special program to help alcoholics. In the Eastmore Program, an alcoholic lives at home but undergoes a two-phase treatment plan. Phase I is an intensive group-therapy program lasting 10 weeks. Phase II is a long-term counseling program lasting 1 year. Eastmore Programs are located in most major cities, and past data gave the following information based on percentages of success and failure collected over a long period of time: The probability that a client will have a relapse in phase I is \(0.27\); the probability that a client will have a relapse in phase II is \(0.23\). However, if a client did not have a relapse in phase I, then the probability that this client will not have a relapse in phase II is \(0.95\). If a client did have a relapse in phase I, then the probability that this client will have a relapse in phase II is \(0.70\). Let \(A\) be the event that a client has a relapse in phase I and \(B\) be the event that a client has a relapse in phase II. Let \(C\) be the event that a client has no relapse in phase \(\mathrm{I}\) and \(D\) be the event that a client has no relapse in phase II. Compute the following: (a) \(P(A), P(B), P(C)\), and \(P(D)\) (b) \(P(B \mid A)\) and \(P(D \mid C)\) (c) \(P(A\) and \(B)\) and \(P(C\) and \(D)\) (d) \(P(A\) or \(B)\) (e) What is the probability that a client will go through both phase I and phase II without a relapse? (f) What is the probability that a client will have a relapse in both phase I and phase II? (g) What is the probability that a client will have a relapse in either phase I or phase II?

Answer questions true or false and give a brief explanation for each answer. Hint: Review the summary of basic probability rules.$$ P\left(A^{c} \text { and } B^{c}\right) \leq 1-P(A) $$

Given \(P(A)=0.7\) and \(P(B)=0.4\) : (a) Can events \(A\) and \(B\) be mutually exclusive? Explain. (b) If \(P(A\) and \(B)=0.2\), compute \(P(A\) or \(B)\).
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