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Chapter 2: Problem 25

In control replication, cells are replicated over a period of two days. Not until mitosis is completed can freshly synthesized DNA be replicated again. Two control mechanisms have been identified - one positive and one negative. Suppose that a replication is observed in three cells. Let \(A\) denote the event that all cells are identified as positive and let \(B\) denote the event that all cells are negative. Describe the sample space graphically and display each of the following events: (a) \(A\) (b) \(B\) (c) \(A \cap B\) (d) \(A \cup B\)

Short Answer

Expert verified
(a) \(A = \{ (P,P,P) \}\); (b) \(B = \{ (N,N,N) \}\); (c) \(A \cap B = \emptyset\); (d) \(A \cup B = \{ (P,P,P), (N,N,N) \}\).

Step by step solution

01

Understand the Events

We are dealing with three cells, each identified as either positive (P) or negative (N). This means for each cell, we have two possible states.
02

Define the Sample Space

The sample space (S) consists of every possible outcome of identifying the states of the three cells. Since each cell can be either P or N, we have:\[ S = \{ (P,P,P), (P,P,N), (P,N,P), (P,N,N), (N,P,P), (N,P,N), (N,N,P), (N,N,N) \} \].There are a total of \(2^3 = 8\) possible outcomes.
03

Define Event A

Event \(A\) is when all cells are identified as positive. Thus, \(A = \{ (P,P,P) \} \).
04

Define Event B

Event \(B\) is when all cells are identified as negative. Thus, \(B = \{ (N,N,N) \} \).
05

Find the Intersection A 鈭� B

The intersection \(A \cap B\) is when all cells are both positive and negative, which is impossible. Thus, \(A \cap B = \emptyset\).
06

Find the Union A 鈭� B

The union \(A \cup B\) is when the event occurs as either all cells positive or all cells negative. Thus, \(A \cup B = \{ (P,P,P), (N,N,N) \} \).

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Key Concepts

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

Sample Space
In probability theory, the **sample space** refers to the set of all possible outcomes of an experiment. For the control replication exercise, we are observing three cells, each of which can be identified as either positive (P) or negative (N). This results in a comprehensive combination of outcomes for all cells.
  • Because there are three cells, and each cell has two possible states, we can calculate the total number of outcomes using the formula for exponential growth: \(2^3 = 8\)
  • Hence, the sample space \( S \) is expressed as: \[ S = \{ (P,P,P), (P,P,N), (P,N,P), (P,N,N), (N,P,P), (N,P,N), (N,N,P), (N,N,N) \} \]
  • This sample space captures every potential state of the cell configuration for this particular setup.
Understanding the sample space is crucial because it sets the stage for evaluating the likelihood of various events defined within this space.
Union of Events
The **union of events** in probability considers at least one of the specified events occurring. When we describe the union of events \(A\) and \(B\), denoted as \( A \cup B \), we include all outcomes that are in either event \(A\) or event \(B\) or in both.
  • For our scenario: Event \(A\) is when all cells are positive: \( A = \{ (P,P,P) \} \)
  • Event \(B\) is when all cells are negative: \( B = \{ (N,N,N) \} \)
  • Therefore, the union is expressed as \( A \cup B = \{ (P,P,P), (N,N,N) \} \)
This union represents the scenarios where all cells are in a homogeneous state, either all positive or all negative.
Intersection of Events
The **intersection of events** refers to the outcomes that are common to two or more events. For two events \( A \) and \( B \), their intersection, denoted by \( A \cap B \), comprises outcomes that occur in both \( A \) and \( B \) simultaneously.
  • Given our exercise, Event \(A\) consists of all positive cells: \( A = \{ (P,P,P) \} \)
  • Event \(B\) consists of all negative cells: \( B = \{ (N,N,N) \} \)
  • Since no outcome can be both all positive and all negative at the same time, the intersection is empty: \( A \cap B = \emptyset \)
An empty intersection means there is no overlap between the two events, illustrating their mutual exclusivity.
Positive and Negative Events
In our control replication context, **positive and negative events** relate to how each cell in our study is identified.
  • A **positive event** (denoted as \( P \)) implies cells are functioning under a positive control mechanism.
  • A **negative event** (denoted as \( N \)) indicates that cells are identified under a negative control mechanism.
  • In practical terms, observing either of these mechanisms across all three cells results in an entirely positive or negative configuration: \( A = \{ (P,P,P) \} \) for positives, and \( B = \{ (N,N,N) \} \) for negatives.
Each configuration stands independently, and the significance of these events plays a critical role in understanding the behavior and characteristics of the cells during the study.

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Most popular questions from this chapter

Semiconductor lasers used in optical storage products require higher power levels for write operations than for read operations. High-power-level operations lower the useful life of the laser. Lasers in products used for backup of higher-speed magnetic disks primarily write, and the probability that the useful life exceeds five years is \(0.95 .\) Lasers that are in products that are used for main storage spend approximately an equal amount of time reading and writing, and the probability that the useful life exceeds five years is \(0.995 .\) Now, \(25 \%\) of the products from a manufacturer are used for backup and \(75 \%\) of the products are used for main storage. Let \(A\) denote the event that a laser's useful life exceeds five years, and let \(B\) denote the event that a laser is in a product that is used for backup. Use a tree diagram to determine the following: (a) \(P(B)\) (b) \(P(A \mid B)\) (c) \(P\left(A \mid B^{\prime}\right)\) (d) \(P(A \cap B)\) (e) \(P\left(A \cap B^{\prime}\right)\) (f) \(P(A)\) (g) What is the probability that the useful life of a laser exceeds five years? (h) What is the probability that a laser that failed before five years came from a product used for backup?

The probability of getting through by telephone to buy concert tickets is \(0.92 .\) For the same event, the probability of accessing the vendor's Web site is \(0.95 .\) Assume that these two ways to buy tickets are independent. What is the probability that someone who tries to buy tickets through the Internet and by phone will obtain tickets?

A player of a video game is confronted with a series of four opponents and an \(80 \%\) probability of defeating each opponent. Assume that the results from opponents are independent (and that when the player is defeated by an opponent the game ends). (a) What is the probability that a player defeats all four opponents in a game? (b) What is the probability that a player defeats at least two opponents in a game? (c) If the game is played three times, what is the probability that the player defeats all four opponents at least once?

In light-dependent photosynthesis, light quality refers to the wavelengths of light that are important. The wavelength of a sample of photosynthetically active radiations (PAR) is measured to the nearest nanometer. The red range is \(675-700 \mathrm{nm}\) and the blue range is \(450-500 \mathrm{nm}\). Let \(A\) denote the event that PAR occurs in the red range and let \(B\) denote the event that PAR occurs in the blue range. Describe the sample space and indicate each of the following events: (a) \(A\) (b) \(B\) (c) \(A \cap B\) (d) \(A \cup B\)

If \(P(A \mid B)=0.4, P(B)=0.8,\) and \(P(A)=0.5,\) are the events \(A\) and \(B\) independent?
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