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Chapter 2: Problem 89

Suppose that \(A\) and \(B\) are two events such that \(P(A)+P(B)<1\). a. What is the smallest possible value for \(P(A \cap B) ?\) b. What is the largest possible value for \(P(A \cap B) ?\)

Short Answer

Expert verified
a. The smallest value is 0. b. The largest value is \(P(A)+P(B)-1\).

Step by step solution

01

Understanding the Intersection of Events

Events A and B are given with their probabilities. We are tasked to find the smallest and largest possible values of their intersection, denoted by \(P(A \cap B)\). This refers to the probability that both events occur simultaneously.
02

Using the Addition Rule for Probabilities

The probability of the union of two events, \(A\) and \(B\), is given by the formula: \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]Given that \(P(A) + P(B) < 1\), we understand that their sum is less than the probability space, leaving room for \(P(A \cap B)\) and non-overlapping probability.
03

Finding the Smallest Possible Value

To minimize \(P(A \cap B)\), consider them as mutually exclusive with no overlap. Set \[P(A \cap B) = 0\]This implies that the smallest possible value is 0 as it represents the scenario where the two events never occur together.
04

Maximizing the Intersection Value

To find the largest possible \(P(A \cap B)\), we need to consider the maximum overlap where almost all of \(P(A) + P(B) < 1\) is the intersection. This value is reached when the probability of the union is simply maximized, and \[P(A \cap B) = P(A) + P(B) - 1\].
05

Conclusion on Bounds for the Intersection

The intersection \(P(A \cap B)\) is bounded by the properties of probabilities and the events given as:- The smallest possible value is 0.- The largest possible value is \(P(A) + P(B) - 1\).

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Key Concepts

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

Intersection of Events
In probability theory, the intersection of events refers to the occurrence of two or more events happening simultaneously. For any two events, \(A\) and \(B\), the intersection is denoted as \(P(A \cap B)\). This means both events \(A\) and \(B\) happen at the same time.

Understanding intersections is crucial because it not only tells us about the possibility of events coinciding but also helps in calculating more complex probabilities. For example, if you are considering the probability that it will rain and you will go outside, you are thinking about the intersection of those two events.
  • The notation \( \cap \) indicates the shared outcome, where both events are realized.
  • If two events are independent, their intersection can be found by multiplying their individual probabilities.
Addition Rule for Probabilities
The Addition Rule for Probabilities is a key concept used to calculate the probability that at least one of several events will occur. For events \(A\) and \(B\), it is stated as:
\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

This formula is essential for combining probabilities and correcting for any overlap between events. When \(A\) and \(B\) can both happen, \(P(A \cap B)\), or their intersection, is subtracted to avoid counting it twice.
  • The term \(P(A \cup B)\) represents the probability that either event \(A\) or event \(B\) or both will occur.
  • If events are mutually exclusive, meaning they cannot both occur at the same time, \(P(A \cap B) = 0\), simplifying the equation to \(P(A \cup B) = P(A) + P(B)\).
Probability Bounds
Probability bounds offer a framework for determining the range within which probabilities of intersections can fall. They help in identifying the limits on \(P(A \cap B)\), based on provided conditions. Generally, we need such bounds to fully grasp the feasibility of occurrences where two events may happen together.

In our exercise, with \(P(A) + P(B) < 1\), the smallest value \(P(A \cap B)\) can take is 0, indicating no overlap, similar to the case of mutually exclusive events. On the other hand, the largest possible value for \(P(A \cap B)\) is \(P(A) + P(B) - 1\), where the total of \(P(A)\) and \(P(B)\) just falls short of absolute certainty.
  • The minimum value 0 is typical when there's no common outcome between events.
  • The upper bound requires understanding of both individual probabilities in relation to their union.
Mutually Exclusive Events
Mutually exclusive events are situations where two events cannot occur at the same time. If \(A\) and \(B\) are mutually exclusive, then \(P(A \cap B) = 0\). This occurs when there's no overlap or shared outcome between \(A\) and \(B\). Consider, for example, flipping a coin. The result can't be both heads and tails for the same flip.

Recognizing mutually exclusive events simplifies many probability calculations, as it implies no intersection. Thus, their combined probability is simply the sum of their individual probabilities.
  • Mutual exclusivity leads directly to the simplification of the addition rule: \(P(A \cup B) = P(A) + P(B)\).
  • This understanding guides problem-solving approaches when estimating probabilities for non-overlapping scenarios.

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Most popular questions from this chapter

In a game, a participant is given three attempts to hit a ball. On each try, she either scores a hit, \(H\), or a miss, \(M\). The game requires that the player must alternate which hand she uses in successive attempts. That is, if she makes her first attempt with her right hand, she must use her left hand for the second attempt and her right hand for the third. Her chance of scoring a hit with her right hand is .7 and with her left hand is \(.4 .\) Assume that the results of successive attempts are independent and that she wins the game if she scores at least two hits in a row. If she makes her first attempt with her right hand, what is the probability that she wins the game?

Consider the following portion of an electric circuit with three relays. Current will flow from point \(a\) to point \(b\) if there is at least one closed path when the relays are activated. The relays may malfunction and not close when activated. Suppose that the relays act independently of one another and close properly when activated, with a probability of \(.9 .\) a. What is the probability that current will flow when the relays are activated? b. Given that current flowed when the relays were activated, what is the probability that relay 1 functioned?

Suppose two balanced coins are tossed and the upper faces are observed. a. List the sample points for this experiment. b. Assign a reasonable probability to each sample point. (Are the sample points equally likely?) c. Let \(A\) denote the event that exactly one head is observed and \(B\) the event that at least one head is observed. List the sample points in both \(A\) and \(B\). d. From your answer to part \((c),\) find \(P(A), P(B), P(A \cap B), P(A \cup B),\) and \(P(\bar{A} \cup B)\)

If \(A\) and \(B\) are mutually exclusive events and \(P(B)>0\), show that $$P(A | A \cup B)=\frac{P(A)}{P(A)+P(B)}$$.

\- Suppose that \(A\) and \(B\) are two events such that \(P(A)=.6\) and \(P(B)=.3\). a. Is it possible that \(P(A \cap B)=.1 ?\) Why or why not? b. What is the smallest possible value for \(P(A \cap B) ?\) c. Is it possible that \(P(A \cap B)=.7 ?\) Why or why not? d. What is the largest possible value for \(P(A \cap B) ?\)
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