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Chapter 7: Problem 50

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a subset of \(B\), then \(P(A) \leq P(B)\).

Short Answer

Expert verified
The statement is true. Since all elements of \(A\) are also elements of \(B\), the probability of \(A\) occurring must be less than or equal to the probability of \(B\) occurring. Mathematically, this is shown as \(P(A) \leq P(B)\).

Step by step solution

01

Understanding subsets and probability

A subset is a set containing elements that are all included in another set. In this problem, \(A\) is a subset of \(B\), which means all elements in \(A\) are also elements in \(B\). Probability is the measure of the likelihood of an event occurring. If \(A\) is an event, \(P(A)\) represents the probability of that event happening. Similarly, \(P(B)\) represents the probability of event \(B\) happening. In this problem, we need to compare the probabilities of events \(A\) and \(B\) when \(A\) is a subset of \(B\).
02

Comparing probabilities of subsets

As \(A\) is a subset of \(B\), we know all elements in \(A\) are also elements in \(B\). Therefore, if there's any probability of an element in \(A\) occurring, this probability must also be included in the probability of \(B\) occurring. This is because \(B\) must account for all its elements and the likelihood of those elements occurring, including those from \(A\). So, the probability of \(A\) occurring must be less than or equal to the probability of \(B\) occurring, since \(B\) contains all the elements of \(A\) as well as its own elements. Mathematically, this means \(P(A) \leq P(B)\).
03

Verifying the statement is true

We have shown that if \(A\) is a subset of \(B\), then \(P(A) \leq P(B)\), based on the fact that all elements in \(A\) are also elements in \(B\). So the given statement is true. To support this, we can provide a brief explanation. Explanation: Since all elements of \(A\) are also elements of \(B\), the probability of \(A\) occurring is either a part of or equal to the probability of \(B\) occurring. Therefore, the probability of event \(A\) must be less than or equal to the probability of event \(B\), which is represented mathematically as \(P(A) \leq P(B)\).

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Most popular questions from this chapter

A halogen desk lamp produced by Luminar was found to be defective. The company has three factories where the lamps are manufactured. The percentage of the total number of halogen desk lamps produced by each factory and the probability that a lamp manufactured by that factory is defective are shown in the accompanying table. What is the probability that the defective lamp was manufactured in factory III? $$ \begin{array}{ccc} \hline & & \text { Probability of } \\ \text { Factory } & \text { Percent of } & \text { Defective } \\ \text { Total Production } & \text { Component } \\ \hline \text { I } & 35 & .015 \\ \hline \text { II } & 35 & .01 \\ \hline \text { III } & 30 & .02 \\ \hline \end{array} $$

In a survey on consumer-spending methods conducted in 2006, the following results were obtained: $$\begin{array}{lccccc} \hline & & & & {\text { Debit/ATM }} & \\ \text { Payment Method } & \text { Checks } & \text { Cash } & \text { Credit cards } & \text { cards } & \text { Other } \\ \hline \text { Transactions, \% } & 37 & 14 & 25 & 15 & 9 \\ \hline \end{array}$$ If a transaction tracked in this survey is selected at random, what is the probability that the transaction was paid for a. With a credit card or with a debit/ATM card? b. With cash or some method other than with a check, a credit card, or a debit/ATM card?

Let \(E\) be any cvent in a sample space \(S .\) a. Are \(E\) and \(S\) independent? Explain your answer. b. Are \(E\) and \(\varnothing\) independent? Explain your answer.

Suppose the probability that Bill can solve a problem is \(p_{1}\) and the probability that Mike can solve it is \(p_{2}\). Show that the probability that Bill and Mike working independently can solve the problem is \(p_{1}+p_{2}-p_{1} p_{2}\).

Before being allowed to enter a maximum-security area at a military installation, a person must pass three independent identification tests: a voice-pattern test, a fingerprint test, and a handwriting test. If the reliability of the first test is \(97 \%\), the reliability of the second test is \(98.5 \%\), and that of the third is \(98.5 \%\), what is the probability that this security system will allow an improperly identified person to enter the maximumsecurity area?enter the maximumsecurity area?
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