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Chapter 4: Problem 100

Given that \(A\) and \(B\) are two mutually exclusive events, find \(P(A\) or \(B\) ) for the following. a. \(P(A)=.38\) and \(P(B)=.59\) b. \(P(A)=.15\) and \(P(B)=.23\)

Short Answer

Expert verified
The probability of either event A or B occurring for given probabilities are 0.97 and 0.38 for the cases a and b respectively.

Step by step solution

01

Calculating the Probability of Either A or B for case a

Let's begin with the first set of given probabilities, which are \(P(A) = 0.38\) and \(P(B) = 0.59\). Given that the two events are mutually exclusive, we can calculate the probability of either event A or event B occurring simply by adding the probabilities of each event. Therefore, \(P(A \, or \, B) = P(A) + P(B) = 0.38 + 0.59 = 0.97\)
02

Calculating the Probability of Either A or B for case b

Now let's do the same for the second set of given probabilities, which are \(P(A) = 0.15\) and \(P(B) = 0.23\). Given that the two events are mutually exclusive, we can calculate the probability of either event A or event B occurring simply by adding the probabilities of each event. Therefore, \(P(A \, or \, B) = P(A) + P(B) = 0.15 + 0.23 = 0.38\)

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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 theory, **mutually exclusive events** are events that cannot occur at the same time. This means that if one event happens, the other cannot. For example, when tossing a coin, you cannot get both heads and tails at the same time; these outcomes are mutually exclusive.
Understanding mutually exclusive events is crucial because it helps us recognize scenarios where outcomes do not overlap. This concept often simplifies the process of calculating the probability of either event occurring.
  • If you know events are mutually exclusive, you can safely add their probabilities to find the probability of either event occurring.
This characteristic is an essential building block for getting comfortable with more complex probability problems.
Probability Addition Rule
The **Probability Addition Rule** is a principle that helps determine the probability of one or more events happening. When dealing with mutually exclusive events, this rule becomes straightforward. The probability of either event A or event B happening is simply the sum of their individual probabilities.
Let's break it down:
  • If two events, A and B, are mutually exclusive, then we can use the formula: \( P(A \ or \ B) = P(A) + P(B) \)
This rule reflects our intuitive understanding that because mutually exclusive events cannot happen together, their combined probability is just about them occurring separately.
In the original exercise, we used this rule to find that:
  • For part a, with \(P(A) = 0.38\) and \(P(B) = 0.59\), \(P(A \ or \ B) = 0.97\)
  • For part b, with \(P(A) = 0.15\) and \(P(B) = 0.23\), \(P(A \ or \ B) = 0.38\)
This simple addition of probabilities works only because the events are mutually exclusive.
Event Probability
**Event Probability** refers to the likelihood that a particular event will occur within a specific context. If the event has already been defined and its probability determined, it allows you to make predictions about the outcome.
When working with probabilities, understanding the specific scenario determines how to approach calculations. For events A and B in the original exercise, the probabilities were given as known values. This provides a clear path to solving the problem using mathematic rules like the addition rule.
  • Each event鈥檚 probability represents a slice of certainty out of a whole (1 or 100%).
  • Probabilities are always between 0 and 1, inclusive, where 0 means the event will not happen, and 1 means it certainly will.
By establishing and knowing the exact probabilities, we can get a good estimate of what to expect when events themselves or their combinations are considered.

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Most popular questions from this chapter

An insurance company has information that \(93 \%\) of its auto policy holders carry collision coverage or uninsured motorist coverage on their policies. Eighty percent of the policy holders carry collision coverage, and \(60 \%\) have uninsured motorist coverage. a. What percentage of these policy holders carry both collision and uninsured motorist coverage? b. What percentage of these policy holders carry neither collision nor uninsured motorist coverage? c. What percentage of these policy holders carry collision but not uninsured motorist coverage?

How many different outcomes are possible for four rolls of a die?

Given that \(P(A \mid B)=.44\) and \(P(A\) and \(B)=.33\), find \(P(B)\).

A random sample of 250 adults was taken, and they were asked whether they prefer watching sports or opera on television. The following table gives the two-way classification of these adults. $$ \begin{array}{lcc} \hline & \begin{array}{c} \text { Prefer Watching } \\ \text { Sports } \end{array} & \begin{array}{c} \text { Prefer Watching } \\ \text { Opera } \end{array} \\ \hline \text { Male } & 96 & 24 \\ \text { Female } & 45 & 85 \\ \hline \end{array} $$ a. If one adult is selected at random from this group, find the probability that this adult i. prefers watching opera ii. is a male iii. prefers watching sports given that the adult is a female iv. is a male given that he prefers watching sports \(\mathrm{v} .\) is a female and prefers watching opera vi. prefers watching sports or is a male b. Are the events "female" and "prefers watching sports" independent? Are they mutually exclusive? Explain why or why not.

Given that \(A, B\), and \(C\) are three independent events, find their joint probability for the following. a. \(P(A)=.81, \quad P(B)=.49\), and \(P(C)=.36\) b. \(P(A)=.02, \quad P(B)=.03\), and \(P(C)=.05\)
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