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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 40 percent of the students in a large population are freshmen, 30 percent are sophomores, 20 percent are juniors, and 10 percent are seniors .Suppose that10 students are selected at random from the population, and let X1, X2, X3, X4 denote, respectively, the numbers of freshmen, sophomores, juniors, and seniors that are obtained.

a. Determine 蟻(Xi, Xj ) for each pair of values i and j (i< j ).

b. For what values of i and j (i<j ) is 蟻(Xi, Xj ) most negative?

c. For what values of i and j (i<j ) is 蟻(Xi, Xj ) closest to 0?

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.

For every collection of events\({A_i}\left( {i \in I} \right)\), show that

\({\left( {\bigcup\limits_{i \in I} {{A_i}} } \right)^c} = \bigcap\limits_{i \in I} {{A_i}^c} \)and\({\left( {\bigcap\limits_{i \in I} {{A_i}} } \right)^c} = \bigcup\limits_{i \in I} {{A_i}^c} \).

A simplified model of the human blood-type systemhas four blood types: A, B, AB, and O. There are twoantigens, anti-Aand anti-B, that react with a person鈥檚blood in different ways depending on the blood type. Anti-A reacts with blood types Aand AB, but not with B and O. Anti-B reacts with blood types B and AB, but not with A and O. Suppose that a person鈥檚 blood is sampled andtested with the two antigens. Let A be the event that theblood reacts with anti-A, and let B be the event that itreacts with anti-B. Classify the person鈥檚 blood type usingthe events A, B, and their complements.

If A, B, and D are three events such that,\({\rm P}\left( {{\rm A} \cup {\rm B} \cup D} \right) = 0.7\)what is the value of\({\rm P}\left( {{{\rm A}^c} \cap {{\rm B}^c} \cap {D^c}} \right)\)?
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