Problem 25 - Calculus and its applications

Question

A dart is thrown at a number line in such a way that it always lands in [0,10] . Let \(x\) represent the number the dart hits. Suppose the probability density function for \(x\) is given by \(f(x)=\frac{1}{50} x, \quad\) for \(0 \leq x \leq 10\) Find \(P(2 \leq x \leq 6),\) the probability that the dart lands in [2,6]

Step-by-Step Solution

Answer

The probability that the dart lands between 2 and 6 is 0.32.

Step 1: Understand the Function and Interval

The probability density function (pdf) given is \( f(x) = \frac{1}{50} x \) for the interval \( 0 \leq x \leq 10 \). We need to find the probability of \( x \) falling between 2 and 6, i.e., \( P(2 \leq x \leq 6) \).

Step 2: Set Up the Integral

The probability \( P(a \leq x \leq b) \) is calculated by integrating the pdf from \( a \) to \( b \). For this problem, we need to set up the integral from 2 to 6 of the function \( f(x) = \frac{1}{50} x \). This is expressed as \( \int_{2}^{6} \frac{1}{50} x \, dx \).

Step 3: Integrate the Function

Integrate the function \( \frac{1}{50} x \) with respect to \( x \). The integral of \( \frac{1}{50} x \) is \( \frac{1}{100} x^2 \). Thus, the solution to the integral \( \int \frac{1}{50} x \, dx \) is \( \frac{1}{100} x^2 + C \), where \( C \) is the constant of integration.

Step 4: Evaluate the Integral from 2 to 6

Evaluate \( \left[ \frac{1}{100} x^2 \right]_{2}^{6} \), which means plug in the upper limit 6 and the lower limit 2 into \( \frac{1}{100} x^2 \) and find the difference. Calculate \( \frac{1}{100} (6^2) - \frac{1}{100} (2^2) \).

Step 5: Simplify the Results

Calculate \( \frac{1}{100} (36) - \frac{1}{100} (4) = \frac{36}{100} - \frac{4}{100} = \frac{32}{100} = 0.32 \). This gives us the probability that the dart lands between 2 and 6.

Key Concepts

Integral Calculus

Integral Calculus is a fundamental branch of mathematics that deals with the accumulation of quantities. By understanding how to calculate areas, volumes, and other quantities that accumulate over intervals, integral calculus is crucial for solving problems involving continuous probability distributions. In our exercise, the function describing the scenario is the probability density function (pdf) given by:

• \( f(x) = \frac{1}{50} x \)

To find the probability that the dart lands between specific values, we use integral calculus. This involves setting up and calculating the definite integral of the pdf over the desired interval. In the example provided, we need to calculate the probability of the dart landing between 2 and 6 on the x-axis. This requires setting up the integral as follows:

• \( \int_{2}^{6} \frac{1}{50} x \, dx \)

Then, integrating the function over this interval gives the desired probability. The integration results in a new function, and by evaluating this integrated function at the boundaries of our interval (6 and 2), we determine the accumulated probability between these points. This method transforms a continuous function into a tangible probability value.

Limits of Integration

Understanding limits of integration is crucial when working with integrals, especially in the context of probability. Limits of integration define the interval over which a function is integrated. In simpler terms, they mark the starting and ending points of the region under the curve that you are interested in.For probability density functions, these limits specify the range over which probabilities are calculated. In our dart-throwing scenario, to find the probability that the dart lands between specific values, we integrate the probability density function within the appropriate limits:

• Lower limit: 2
• Upper limit: 6

These bounds form the interval \([2,6]\) on the number line, capturing where the dart could possibly land within that subrange of 0 to 10. By integrating from 2 to 6, the result of the integral reflects the total probability associated with the interval. This is essential, as probabilities for a continuous distribution are found from the area under the curve between these limits.

Continuous Probability Distribution

A continuous probability distribution describes outcomes that can take on any value within a certain range. Unlike discrete distributions, where outcomes are separate and countable, continuous distributions deal with outcomes that are not individually countable, like measuring the exact position on a number line.The given exercise features a continuous probability distribution represented by a linear probability density function:

• \( f(x) = \frac{1}{50} x \)
• Domain: \(0 \leq x \leq 10\)

This distribution implies that any value between 0 and 10 is possible for the dart's landing point, with varying probabilities.The probability density function (pdf) dictates how probabilities are distributed over the interval. Although an individual point like 5 has a probability of zero in a continuous distribution, entire ranges such as \([2,6]\) do account for probabilities. To find the probability of landing within a particular range, we calculate the integral of the pdf over that interval, as seen in the solution.Continuous probability distributions, thus, help model real-world scenarios where outcomes are not simply black or white, allowing for a more nuanced approach to predicting outcomes.