Problem 9 - Calculus for Biology and Medicine

Question

In Problems \(9-12\), assume that $$ \Omega=\\{1,2,3,4,5\\} $$ \(P(1)=0.1, P(2)=0.2\), and \(P(3)=P(4)=0.05 .\) Furthermore, assume that \(A=\\{1,3,5\\}\) and \(B=\\{2,3,4\\}\). Find \(P(5)\)

Step-by-Step Solution

Answer

\(P(5) = 0.6\)

Step 1: Understanding Total Probability

The total probability of all outcomes in the sample space, \( \Omega \, = \, \{1, 2, 3, 4, 5\} \), must sum to 1. That is, \( P(1) + P(2) + P(3) + P(4) + P(5) = 1 \).

Step 2: Substitute Known Probabilities

Substitute the known probabilities into the equation from Step 1:\[ P(1) + P(2) + P(3) + P(4) + P(5) = 1 \]\[ 0.1 + 0.2 + 0.05 + 0.05 + P(5) = 1 \].

Step 3: Simplify the Equation

Add the known probabilities on the left side of the equation:\[ 0.4 + P(5) = 1 \].

Step 4: Solve for \(P(5)\)

To find \(P(5)\), subtract 0.4 from both sides of the equation:\[ P(5) = 1 - 0.4 \].

Step 5: Calculate the value of \(P(5)\)

Perform the subtraction to get:\[ P(5) = 0.6 \].

Key Concepts

Sample Space

In probability theory, the sample space of an experiment is a complete set that contains all possible outcomes. Think of it as a list of everything that might happen when you perform an action.
For example, if you roll a six-sided die, then the sample space is the set \( \{1, 2, 3, 4, 5, 6\} \). Every number represents a possible result.
In the context of the exercise we have, the sample space is given as \( \Omega = \{1, 2, 3, 4, 5\} \). Each number from 1 to 5 represents an outcome that can occur.

• 1, 2, 3, 4, and 5 are each possible outcomes.
• This set includes everything that can happen.

Understanding the sample space is crucial because probabilities are calculated based on these possible outcomes.
It’s the foundation of any probability calculation, including the event sets \(A\) and \(B\) in the exercise, which are subsets of this complete sample space. In any sample space, the sum of probabilities of all individual outcomes always equals 1, reflecting certain certainty that one of these outcomes will occur.

Total Probability

Total probability refers to the sum of probabilities of all possible outcomes within the sample space. Imagine you have a complete list of possible events; the probability of one of these happening is assured, summed up to a perfect score of 1.
In mathematical terms, for a sample space \( \Omega \) with outcomes \( \{1, 2, 3, 4, 5\} \), the equation is:

• \( P(1) + P(2) + P(3) + P(4) + P(5) = 1 \)

The given exercise demonstrates this by assigning specific probabilities to some of these outcomes and then finding the missing probability to ensure that the total is equal to 1.
In our given problem, after knowing that \( P(1) = 0.1 \), \( P(2) = 0.2 \), and \( P(3) = P(4) = 0.05 \), the task was to find \( P(5) \) such that they all add up to 1. By substituting the known values and solving the equation, we find \( P(5) = 0.6 \). Understanding this concept lets us see how every part contributes to a whole.

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. In probability, we often describe events as sets of outcomes within a sample space.
Each event in probability can be seen as a subset of the sample space.
Consider the sets \(A = \{1, 3, 5\}\) and \(B = \{2, 3, 4\}\) from the exercise. These are subsets of the sample space \(\Omega = \{1, 2, 3, 4, 5\}\).

• Set \(A\) contains outcomes 1, 3, and 5.
• Set \(B\) holds outcomes 2, 3, and 4.

Set theory helps us understand how these subsets interact and the probability of these interactions.
Common operations include unions and intersections:

• The **union** of two sets \(A\) and \(B\) consists of all outcomes in either set.
• The **intersection** comprises outcomes that appear in both sets.

This mathematical foundation allows us to describe and solve complex problems involving probability by using operations on sets to find probabilities of combined or overlapping events.