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Chapter 1: Q3E (page 25)

If two balanced dice are rolled, what is the probability that the difference between the two numbers that appear will be less than 3?

Short Answer

Expert verified

The probability that the difference between two numbers that appear will be less than 3 is 0.666667

Step by step solution

01

Given information

Here we rolled two balanced dice

02

Define events and sample space

Let S represent the sample space of the experiments.

\(S = \left\{ \begin{aligned}{l}\left( {1,1} \right),\left( {1,2} \right),\left( {1,3} \right),\left( {1,4} \right),\left( {1,5} \right),\left( {1,6} \right)\\\,\left( {2,1} \right),\left( {2,2} \right),\left( {2,3} \right),\left( {2,4} \right),\left( {2,5} \right),\left( {2,6} \right)\\\,\left( {3,1} \right),\left( {3,2} \right),\left( {3,3} \right),\left( {3,4} \right),\left( {3,5} \right),\left( {3,6} \right)\\\,\left( {4,1} \right),\left( {4,2} \right),\left( {4,3} \right),\left( {4,4} \right),\left( {4,5} \right),\left( {4,6} \right)\\\,\left( {5,1} \right),\left( {5,2} \right),\left( {5,3} \right),\left( {5,4} \right),\left( {5,5} \right),\left( {5,6} \right)\\\,\,\left( {6,1} \right),\left( {6,2} \right),\left( {6,3} \right),\left( {6,4} \right),\left( {6,5} \right),\left( {6,6} \right)\end{aligned} \right\}\)

In the sample space, the total possible outcomes are 36

Let A be the event that represents the difference between two numbers that appear will be less than 3

\(A = \left\{ \begin{aligned}{l}\left( {1,1} \right),\left( {1,2} \right),\left( {1,3} \right),\left( {2,1} \right),\left( {2,2} \right),\left( {2,3} \right),\left( {2,4} \right),\,\left( {3,1} \right),\left( {3,2} \right),\\\left( {3,3} \right),\left( {3,4} \right),\left( {3,5} \right),\left( {4,2} \right),\left( {4,3} \right),\left( {4,4} \right),\left( {4,5} \right),\left( {4,6} \right),\left( {5,3} \right),\\\left( {5,4} \right),\left( {5,5} \right),\left( {5,6} \right)\,,\left( {6,4} \right),\left( {6,5} \right),\left( {6,6} \right)\end{aligned} \right\}\)

The number of favourable outcomes for this experiment is 24

\(n\left( A \right) = 24\)

03

Calculate the probability

The probability of the difference between two numbers that appear to be less than 3 is

\(\begin{aligned}{}\Pr \left( A \right) &= \frac{{{\bf{n}}\left( {\bf{A}} \right)}}{{{\bf{n}}\left( {\bf{S}} \right)}}\\ &= \frac{{24}}{{36}}\\ &= 0.666667\end{aligned}\)

The probability of the difference between two numbers that appear will be less than 3 is 0.666667

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Most popular questions from this chapter

Suppose that a school band contains 10 students from the freshman class, 20 students from the sophomore class, 30 students from the junior class, and 40 students from the senior class. If 15 students are selected at random from the band, what is the probability that at least one student will be selected from each of the four classes Hint: First determine the probability that at least one of the four classes will not be represented in the selection.

If k people are seated in a random manner in a row containing n seats (n > k), what is the probability that the people will occupy k adjacent seats in the row?

Suppose that one card is to be selected from a deck of 20 cards that contains 10 red cards numbered from 1 to 10 and 10 blue cards numbered from 1 to 10. Let A be the event that a card with an even number is selected, let B be the event that a blue card is selected, and let C be the event that a card with a number less than 5 is selected. Describe the sample space S and describe each of the following events both in words and as subsets of S:

a. \({\bf{A}} \cap {\bf{B}} \cap {\bf{C}}\)

b. \({\bf{B}} \cap {{\bf{C}}^{\bf{C}}}\)

c. \({\bf{A}} \cup {\bf{B}} \cup {\bf{C}}\)

d. \({\bf{A}} \cap {\bf{(B}} \cup {\bf{C)}}\)

e. \({{\bf{A}}^{\bf{c}}} \cap {{\bf{B}}^{\bf{c}}} \cap {{\bf{C}}^{\bf{c}}}\)

A box contains 24 light bulbs, of which four are defective. If a person selects four bulbs from the box at random, without replacement, what is the probability that all four bulbs will be defective?

Consider two events A and B with Pr(A) = 0.4 and Pr(B) = 0.7. Determine the maximum and minimum possible values of \(Pr\left( {A \cap B} \right)\) and the conditions under which each of these values is attained.
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