Problem 40 - Calculus for Biology and Medicine

Question

Suppose that you pick five numbers at random from the interval \((0,1)\). Assume that the numbers are independent. What is the probability that all numbers are greater than \(0.7 ?\)

Step-by-Step Solution

Answer

The probability is 0.00243.

Step 1: Define the Probability of a Single Event

We start by determining the probability that a single randomly chosen number from the interval \((0,1)\) is greater than 0.7. Since the interval is \((0,1)\), any specific interval has a probability equal to the length of the interval. Therefore, the probability that one number is greater than 0.7 is \(1 - 0.7 = 0.3\).

Step 2: Determine the Probability for All Numbers

Next, we calculate the probability that all five numbers are greater than 0.7. Since the numbers are independent, we multiply the probability of the single event by itself for all numbers. Thus, the probability is given by \( (0.3)^5 \).

Step 3: Calculate the Result

Now, we perform the calculation: \( (0.3)^5 = 0.00243 \). This result is the probability that all five numbers are greater than 0.7.

Key Concepts

Independent Events

In probability theory, the concept of independent events is essential. Independent events are those whose outcomes do not influence each other. This means that the occurrence or non-occurrence of one event does not affect the probability of the other event occurring. For example, when you roll two dice, the result of one die does not affect the result of the other die.
This concept is particularly important when determining the combined probability of multiple events. If events are independent, the probability of all occurring is the product of their individual probabilities. This rule simplifies complex probability calculations, making it easier to handle scenarios involving numerous events.

Probability Calculation

Calculating probability involves determining how likely an event is to occur. It's usually expressed as a number between 0 and 1, where 0 indicates impossibility, and 1 indicates certainty. To calculate the probability of independent events occurring together, we multiply their individual probabilities.

• First, identify the probability of a single event happening. For instance, in the given example, the probability a number is greater than 0.7 in the interval \( (0,1) \) is calculated as \( 1 - 0.7 = 0.3 \).
• Next, multiply the probability for all events. Since we want all five numbers to be greater than 0.7, we calculate \( (0.3)^5 = 0.00243 \).

This process of probability calculation is foundational in understanding how likely a collective group of independent events is to occur.

Interval Selection

Interval selection is a crucial part of problems involving probability theory, especially when continuous probabilities are involved. An interval represents a range of values, such as \( (a, b) \), where any number within this range could be selected.
Selecting an interval, like \( (0,1) \), specifies all possible outcomes within that range. Understanding how to work with intervals helps in determining probabilities of selecting numbers greater than a certain value. For example, calculating how many numbers are greater than 0.7 in the interval involves understanding that the probability is linked directly to the length of the sub-interval, \( (0.7, 1) \).
Grasping how intervals work allows for more precise mathematical analysis in probability problems, as seen when deducing probabilities based on the range of acceptable values.