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Chapter 4: Problem 26

A die is rolled once. What is the probability that a. a number less than 5 is obtained? b. a number 3 to 6 is obtained?

Short Answer

Expert verified
The probability that a dice roll results in a number less than 5 is \(\frac{2}{3}\). Similarly, the probability of obtaining a number between 3 and 6 (inclusive) is also \(\frac{2}{3}\).

Step by step solution

01

Identify the total number of outcomes.

When a die is rolled, the total number of outcomes is 6 because a die has six faces, each with a distinct number from 1 to 6.
02

Calculate the probability for part a.

The favorable outcomes for part a are getting the numbers 1, 2, 3, and 4, making a total of 4 favorable outcomes. Hence, the probability of getting a number less than 5 is calculated by dividing the number of favorable outcomes by the total number of outcomes, which is \(\frac{4}{6}\) or \(\frac{2}{3}\).
03

Calculate the probability for part b.

The favorable outcomes for part b are getting the numbers 3, 4, 5, and 6, making a total of 4 favorable outcomes. Hence, the probability of getting a number between 3 and 6 is calculated by dividing the number of favorable outcomes by the total number of outcomes, which is \(\frac{4}{6}\) or \(\frac{2}{3}\).

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Most popular questions from this chapter

A random sample of 80 lawyers was taken, and they were asked if they are in favor of or against capital punishment. The following table gives the two-way classification of their responses. $$ \begin{array}{lcc} \hline & \begin{array}{c} \text { Favors Capital } \\ \text { Punishment } \end{array} & \begin{array}{c} \text { Opposes Capital } \\ \text { Punishment } \end{array} \\ \hline \text { Male } & 32 & 24 \\ \text { Female } & 13 & 11 \\ \hline \end{array} $$ a. If one lawyer is randomly selected from this group, find the probability that this lawyer i. favors capital punishment ii. is a female iii. opposes capital punishment given that the lawyer is a female iv. is a male given that he favors capital punishment \(\mathrm{v}\). is a female and favors capital punishment vi. opposes capital punishment or is a male b. Are the events "female" and "opposes capital punishment" independent? Are they mutually exclusive? Explain why or why not.

Find \(P(A\) or \(B)\) for the following. a. \(P(A)=.28, \quad P(B)=.39\), and \(P(A\) and \(B)=.08\) b. \(P(A)=.41, \quad P(B)=.27\), and \(P(A\) and \(B)=.19\)

How is the multiplication rule of probability for two dependent events different from the rule for two independent events?

Find the joint probability of \(A\) and \(B\) for the following. a. \(P(A)=.36\) and \(P(B \mid A)=.87\) b. \(P(B)=.53\) and \(P(A \mid B)=.22\)

Given that \(P(A)=.72\) and \(P(A\) and \(B)=.38\), find \(P(B \mid A)\).
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