Problem 26 - Calculus and its applications

Question

In Exercise \(25,\) suppose the dart always lands in \([0,5],\) and the probability density function for \(x\) is given by \(f(x)=\frac{3}{125} x^{2}, \quad\) for \(0 \leq x \leq 5\) Find \(P(1 \leq x \leq 4)\), the probability that the dart lands in [1,4]

Step-by-Step Solution

Answer

The probability that the dart lands in \([1,4]\) is 0.504.

Step 1: Identify the Given Probability Density Function

The probability density function (pdf) provided is \( f(x) = \frac{3}{125} x^{2} \) for \( 0 \leq x \leq 5 \). This function describes how the probability is distributed along the interval \([0,5]\).

Step 2: Set Up the Probability Integral

To find the probability that the dart lands in the interval \([1,4]\), we need to compute the integral of the function \( f(x) = \frac{3}{125} x^{2} \) from \( x = 1 \) to \( x = 4 \). This will give us the total probability within this specific range.

Step 3: Compute the Indefinite Integral

First, find the indefinite integral of \( f(x) \), which is the integral of \( \frac{3}{125} x^{2} \). \[ \int \frac{3}{125} x^{2} \, dx = \frac{3}{125} \cdot \frac{x^{3}}{3} = \frac{1}{125} x^{3} + C \] where \( C \) is the integration constant.

Step 4: Evaluate the Definite Integral from 1 to 4

Next, evaluate the definite integral \( \int_{1}^{4} \frac{3}{125} x^{2} \, dx \) using the result from the indefinite integral: \[ \left[ \frac{1}{125} x^{3} \right]_{1}^{4} = \frac{1}{125}(4^{3}) - \frac{1}{125}(1^{3}) \] This simplifies to \( \frac{1}{125}(64) - \frac{1}{125}(1) \).

Step 5: Simplify the Result

Simplifying the expression: \[ \frac{1}{125}(64) - \frac{1}{125}(1) = \frac{64}{125} - \frac{1}{125} = \frac{63}{125} \] Therefore, \( P(1 \leq x \leq 4) = \frac{63}{125} = 0.504 \).

Key Concepts

Probability

Probability is a measure of how likely an event is to happen. In this exercise, we're looking at the probability density function (pdf) of a dart landing on a number line from 0 to 5. It tells us the likelihood of the dart landing at any given point on this interval. A probability density function must satisfy two conditions:

• The function must be non-negative across its entire range.
• The total area under the curve of the pdf across the interval must equal 1, ensuring all probabilities sum up correctly.

This exercise focuses on finding the probability that the dart lands specifically between 1 and 4. To do this, we find the area under the curve of the function from 1 to 4, which is where the integral comes in.

Definite Integral

A definite integral is a powerful mathematical tool that computes the accumulation of quantities, such as area under a curve. Unlike an indefinite integral, which includes a constant of integration, a definite integral provides the actual value representing the total change or area over a specific interval. In this case, we use the definite integral to find the exact probability that the dart lands between 1 and 4 on the number line. The process involves the following steps:

• We first calculate the indefinite integral of the probability density function, which gives us the antiderivative.
• Next, we evaluate this antiderivative at the boundaries of the interval [1, 4]. This involves plugging in the upper limit, 4, and the lower limit, 1, and then subtracting these values.

This calculation gives the exact area under the curve from x = 1 to x = 4, providing the sought probability.

Indefinite Integral

An indefinite integral, unlike a definite one, finds the general form of antiderivatives of a function and includes a constant of integration. This is because when finding the antiderivative, there are infinite possible functions that differ only by a constant. For our function, integrating \( f(x) = \frac{3}{125} x^2 \) results in \( \int \frac{3}{125} x^2 \, dx = \frac{1}{125} x^3 + C \).The constant \( C \) is part of this general solution. However, when computing definite integrals, this constant cancels out, leaving only the area between the specified bounds. In practical terms, finding the indefinite integral helps obtain the antiderivative formula. This formula is crucial for calculating the value of definite integrals over specific intervals, which in our case, provides the probability of interest.