Problem 37 - Calculus for Biology and Medicine

Question

Suppose that you pick a number at random from the interval \((0,4)\). What is the probability that the first digit after the decimal point is a 3 ?

Step-by-Step Solution

Answer

The probability is 0.1 or 10%.

Step 1: Understanding the Interval

The interval given is \(0, 4\), which means any random number chosen will be greater than 0 and less than 4. These numbers can be represented in decimal form as \(x = a.b\), where \(a\) is the integer part (0 to 3) and \(b\) is the decimal part.

Step 2: Focusing on the Decimal Part

For this exercise, only the first digit after the decimal point is of interest. Let's denote this first digit as \(d_1\) in \(x = a.d_1d_2...\). We need \(d_1 = 3\) regardless of the integer part of the number.

Step 3: Calculating the Range for the First Decimal Digit

The first digit after the decimal point being 3 means:- If \(a = 0\), \(x = 0.3y\);- If \(a = 1\), \(x = 1.3y\);- If \(a = 2\), \(x = 2.3y\);- If \(a = 3\), \(x = 3.3y\);where \(0 \leq y < 1\). In each case, \(d_1\) will be 3 if the number is in the range \( [a.3, a.4) \).

Step 4: Calculating the Favored Interval Length

For each \(a\) (0 to 3), the range of numbers where the first decimal digit is 3 is exactly \(0.1\) units long ([a.3, a.4)). Therefore, for each range of 1 unit length (for example from 0 to 1, or from 1 to 2), the length of numbers having \(d_1 = 3\) is \(0.1\).

Step 5: Determining Total Probability

The total length of the interval \(0, 4\) is 4. Since each length of the sub-interval where \(d_1 = 3\) is \(0.1\) and there are 4 such intervals ([0,1), [1,2), [2,3), [3,4)), the total length for \(d_1 = 3\) is \(4 imes 0.1 = 0.4\). The probability is thus \(\frac{0.4}{4} = 0.1\).

Key Concepts

Random Selection

In the context of probability, random selection involves choosing an element from a set where each element has an equal chance of being chosen. For this exercise, we are randomly selecting a number from the interval \((0,4)\). This means any number can be picked from the range, starting just above 0 up to but not including 4.This selection process doesn't favor any specific number over another within the range, ensuring that each value has an equal opportunity of being selected. Random selection in this case is continuous, meaning it includes every possible number between 0 and 4.- For instance, any number like 1.5, 2.7, or 3.9 could be chosen.- Random selection in probability ensures fairness in picking, which is crucial for accurate calculation of chances and outcomes.

Decimal Interval

To better understand the probability question, it’s important to comprehend what a decimal interval in this context is. A decimal interval refers to a specific range of numbers, defined by its starting point and ending point within which the random selection occurs.For our problem, numbers are chosen from the interval \((0,4)\)\. This is a continuous interval, meaning it includes all real numbers from just above 0 to just below 4. When we focus on the decimal numbers, such as 0.1 or 3.7, it becomes vital to understand how these decimals form intervals themselves.For example, consider the scenario within each whole number part:- Numbers between 0.3 and 0.4 (inclusive of 0.3 but not 0.4) form a specific decimal interval.- Each interval like 0.3 to 0.4 has a length of 0.1.- This pattern repeats for each integer range: [0.3, 0.4), [1.3, 1.4), [2.3, 2.4), and [3.3, 3.4). Thus, understanding decimal intervals helps us identify ranges where specific conditions, like a particular first decimal digit, are met.

Probability Calculation

Probability calculation involves determining the likelihood of an event occurring. It is represented in various ways, but commonly as a fraction, decimal, or percentage.For our exercise, the event is specifically about the first digit after the decimal point being '3'. We calculate this probability by first understanding the interval and then analyzing where each condition is satisfied.Here's how it's broken down:- We know that the interval is \((0,4)\)\.- We calculate where the first decimal digit equals 3, which happens in the intervals [0.3, 0.4), [1.3, 1.4), [2.3, 2.4), [3.3, 3.4).- The length of each of these intervals is 0.1, aligning with where our condition is met.By multiplying the length of one favorable interval (0.1) by the number of such intervals (4), we find that the total length of the interval meeting our condition is 0.4. Finally, to get the probability, we divide the favorable condition length by the total interval length, \[ \frac{0.4}{4} = 0.1 \]\. Therefore, the probability that a randomly picked number will have '3' as its first decimal digit is 0.1, or 10%.