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 0.4.

Step 1: Understanding the Interval

The range \(0,4\) indicates we are picking any number between 0 and 4, not including 0 but including any number slightly less than 4.

Step 2: Identifying the Relevant Range

For a number to have 3 as the first digit after the decimal in the range \(0,1\), it should be between 0.30 and 0.40. Similarly, for the ranges \(1,2\), \(2,3\), and \(3,4\), the numbers must lie between 1.30 and 1.40, between 2.30 and 2.40, and between 3.30 and 3.40, respectively.

Step 3: Calculating Probability for Each Segment

Each segment, \(0.3 \, ext{to} \, 0.4\), \(1.3 \, ext{to} \, 1.4\), \(2.3 \, ext{to} \, 2.4\), and \(3.3 \, ext{to} \, 3.4\), has a length of 0.1. The probability of picking a number in each segment is the length of the segment, which is 0.1.

Step 4: Summing Probabilities

Since there are four such intervals, the total probability is the sum of probabilities of each interval: \(0.1 + 0.1 + 0.1 + 0.1 = 0.4\).

Key Concepts

Intervals

In probability, an interval refers to the range of numbers from which you can choose your possible outcomes. Understanding intervals is crucial as it sets the boundaries within which random numbers are selected.
For the interval \(0, 4\), the numbers you pick can include any decimal within this range, such as 1.23 or 3.78.
Importantly, this interval is open on the left and closed on the right, meaning 0 is not included but values arbitrarily close to 4 are.
This nuanced understanding of an interval is foundational for probability calculations, as it determines the space in which events can occur.

Decimal Points

Decimal points play a significant role in probability, especially when identifying specific regions within intervals.
In our problem, we focus on the first digit after the decimal point for numbers within the interval \(0, 4\).
This specific focus helps to narrow down segments of interest, such as numbers ranging from 0.30 to 0.40 for numbers slightly greater than 0. Within each cycle of whole numbers (1, 2, 3, etc.), these decimal ranges determine specific outcomes.
Thus, understanding and accurately identifying the effect of decimal points can simplify and target probability calculations.

Probability calculation

Probability calculation involves determining the likelihood of a particular event occurring.
In our scenario, we calculate the probability of randomly picking a number in specific segments of our interval \(0, 4\) where the first digit after the decimal is 3.
Each subinterval, like \(0.30 - 0.40\) or \(2.30 - 2.40\), has the same length of 0.1, representing the probability of selecting a number from that segment.
Combining probabilities from all these equal segments gives the total probability of the desired outcome. In this exercise, we summed four decimal segments yielding probabilities, resulting in a total probability of 0.4.