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Chapter 11: Problem 26

A single die is rolled. Find the probability of rolling an odd number or a number less than 4 .

Short Answer

Expert verified
The probability of rolling an odd number or a number less than 4 is \( \frac{2}{3} \).

Step by step solution

01

Identify the sample space

The sample space related to the experiment of rolling a die is denoted as S = \{1, 2, 3, 4, 5, 6\}. This is because a die has six sides, each side having a distinct number from 1 to 6.
02

Define the events

Let's denote the event of rolling an odd number as E1 = \{1, 3, 5\} and the event of rolling a number less than 4 as E2 = \{1, 2, 3\}.
03

Calculate individual probabilities

The probability of an event is calculated as the ratio of the number of outcomes in the event set to the number of outcomes in the sample space. So, P(E1) = \(\frac{3}{6} = 0.5\) and P(E2) = \(\frac{3}{6} = 0.5\).
04

Apply the rule of sum

Since E1 and E2 are not mutually exclusive (they have one or more common outcomes), the rule of sum (also known as the addition rule) for the probability of the union of two events should be applied. According to this rule, P(A or B) = P(A) + P(B) - P(A and B).
05

Calculate the probability of the union of events

Let's denote A and B as E1 and E2, respectively. The event (A and B) corresponds to the set of outcomes which are common to both E1 and E2, that is, \{1, 3\}. Hence, P(A and B) = \(\frac{2}{6} = \frac{1}{3}\). Therefore, by applying the rule of sum, P(E1 or E2) = 0.5 + 0.5 - \(\frac{1}{3} = \frac{2}{3}\).

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