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Chapter 2: Problem 5

Find the probability of the union of the events \(a

Short Answer

Expert verified
The probability of the union of the events is \(\frac{17}{24}\).

Step by step solution

01

- Understanding independent events

The first thing to remember is that the word 'independent' means that the result of one event does not affect the result of the other event. This is very important for understanding how to solve this problem.
02

- Using the formula for the union of two independent events

To find the probability of the union of two independent events, we can use the formula \(P(A \cup B) = P(A) + P(B) - P(A)P(B)\). Please note that this formula applies to independent events, which is known to be the case in this exercise. Here, we treat our events \(a<X_{1}<b, -\infty<X_{2}<\infty\) and \(-\infty<X_{1}<\infty, c<X_{2}<d\) as A and B, respectively.
03

- Substituting the given probabilities into the formula

This exercise has provided us with the individual probabilities \(P(A) = \frac{2}{3}\) and \(P(B) = \frac{5}{8}\). We substitute these values into the union formula.
04

- Calculating the union probability

Using the values, the union of these two events will then be:\(P(A \cup B) = P(A) + P(B) - P(A)P(B) = \frac{2}{3} + \frac{5}{8} - \frac{2}{3} \cdot \frac{5}{8}\). We need to calculate this expression to find the particular value.
05

- Simplifying the expression

After simplifying, the final probability value will be: \(P(A \cup B) = \frac{16}{24} + \frac{15}{24} - \frac{60}{192} = \frac{31}{24} - \frac{15}{64} = \frac{136}{192} = \frac{17}{24}\).

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Most popular questions from this chapter

Let \(X_{1}\) and \(X_{2}\) be independent random variables with nonzero variances. Find the correlation coefficient of \(Y=X_{1} X_{2}\) and \(X_{1}\) in terms of the means and variances of \(X_{1}\) and \(X_{2}\).

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