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Chapter 4: Problem 3

What is the probability of (a) an event \(A\) that is certain to occur? (b) an event \(B\) that is impossible?

Short Answer

Expert verified
(a) Probability of a certain event (A) is 1. (b) Probability of an impossible event (B) is 0.

Step by step solution

01

Understanding Probability Range

Probability is a measure of the likelihood of an event occurring and it ranges from 0 to 1. An event that is certain to happen has a probability of 1, while an event that is impossible has a probability of 0.
02

Analyzing Event A (Certain Event)

Since event A is certain to occur, it means that there is no doubt that this event will happen. Therefore, the probability of an event A that is certain to occur is 1.
03

Analyzing Event B (Impossible Event)

Since event B is impossible, it indicates that there is no chance of this event occurring. Therefore, the probability of an event B that is impossible is 0.

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Key Concepts

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

Certain Event
A certain event in probability theory is one that is guaranteed to happen. It's like knowing that the sun will rise tomorrow 鈥� there's no doubt about it. Thus, the probability of a certain event is always 1. This 1 represents 100% certainty. In our daily lives, "certain events" are often things we can safely assume will occur without exception.
  • For example, flipping a coin and getting either heads or tails is a certain event when considering both outcomes.
  • When rolling a standard six-sided die, rolling a number between 1 and 6 is a certain event, as the die cannot land on any number outside this range.
Understanding certain events can help frame our expectations in both simple and complex scenarios. It's a foundational concept in probability that assures us that we can predict some events with complete confidence.
Impossible Event
In the realm of probability, an impossible event has a probability of 0. This indicates that the event cannot occur under any circumstances. For instance, rolling a standard six-sided die and expecting to land on the number 7 is an impossible event. This concept is the direct opposite of a certain event and combines with it to define the entire spectrum of probability. Recognizing impossible events helps us understand the true limits and constraints within a given scenario.
  • Consider trying to flip a fair coin and expecting it to land on its edge: it's nearly impossible, making its probability essentially 0.
  • For picking a red card from a deck that has only black cards, the probability of picking red is also 0, categorizing this as an impossible event.
Impossible events serve as vital markers in our understanding of what is and isn't feasible within defined circumstances.
Probability Range
The probability range is a crucial concept in probability theory; it defines the spectrum within which probabilities lie. The range is between 0 and 1, inclusive. This means that any event has a probability within these bounds.
  • A probability of 0 signals that the event is impossible.
  • A probability of 1 affirms that the event is certain.
  • Probabilities between 0 and 1 reflect the varying likelihood of events occurring. For instance, a probability of 0.5 indicates that an event is just as likely to happen as it is not to happen.
This range helps in quantifying uncertainty and understanding the likelihood of various outcomes. It is visually represented on a probability line, which is a useful tool in illustrating where events fall in terms of their likelihood.

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Most popular questions from this chapter

You roll two fair dice, a green one and a red one. (a) Are the outcomes on the dice independent? (b) Find \(P(5\) on green die and 3 on red die). (c) Find \(P(3\) on green die and 5 on red die). (d) Find \(P((5\) on green die and 3 on red die) or \((3\) on green die and 5 on red die \()\).

John runs a computer software store. Yesterday he counted 127 people who walked by his store, 58 of whom came into the store. Of the 58 , only 25 bought something in the store. (a) Estimate the probability that a person who walks by the store will enter the store. (b) Estimate the probability that a person who walks into the store will buy something. (c) Estimate the probability that a person who walks by the store will come in and buy something. (d) Estimate the probability that a person who comes into the store will buy nothing.

Sometimes probability statements are expressed in terms of odds. The odds in favor of an event \(A\) is the ratio \(\frac{P(A)}{P(n o t A)}=\frac{P(A)}{P\left(A^{c}\right)}\). For instance, if \(P(A)=0.60\), then \(P\left(A^{c}\right)=0.40\) and the odds in favor of \(A\) are \(\frac{0.60}{0.40}=\frac{6}{4}=\frac{3}{2}\), written as 3 to 2 or \(3: 2 .\) (a) Show that if we are given the odds in favor of event \(A\) as \(n: m\), the probability of event \(A\) is given by \(P(A)=\frac{n}{n+m} \cdot\) Hint \(:\) Solve the equation \(\frac{n}{m}=\frac{P(A)}{1-P(A)}\) for \(P(\bar{A})\). (b) A telemarketing supervisor tells a new worker that the odds of making a sale on a single call are 2 to \(15 .\) What is the probability of a successful call? (c) A sports announcer says that the odds a basketball player will make a free throw shot are 3 to \(5 .\) What is the probability the player will make the shot?

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 \(\mathrm{I}\) 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 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?

You draw two cards from a standard deck of 52 cards, but before you draw the second card, you put the first one back and reshuffle the deck. (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.
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