Question

In a certain population, \(85 \%\) of the people have Rh-positive blood. Suppose that two people from this population get married. What is the probability that they are both Rh-negative, thus making it inevitable that their children will be Rh-negative?

Short answer

Answer: 2.25%

Step-by-step solution

Find the Probability of Rh-negative Blood

First, we need to find the probability of a person having Rh-negative blood. We are given the probability of a person having Rh-positive blood as \(85 \%\). Using the complement rule, we can calculate the probability of a person having Rh-negative blood as follows: \(P(\text{Rh-negative}) = 1 - P(\text{Rh-positive})\) \(P(\text{Rh-negative}) = 1 - 0.85\) \(P(\text{Rh-negative}) = 0.15\) So, the probability of a person having Rh-negative blood is \(15 \%\).

Find the Probability of Both People in the Marriage Being Rh-negative

Now, we need to find the probability of both people in the marriage having Rh-negative blood. Since the two people are independent, we can multiply the probabilities of each of them having Rh-negative blood to get the probability of both having Rh-negative blood: \(P(\text{Both Rh-negative}) = P(\text{Rh-negative Person 1}) \times P(\text{Rh-negative Person 2})\) Since both probabilities are the same, we have: \(P(\text{Both Rh-negative}) = (0.15) \times (0.15)\) \(P(\text{Both Rh-negative}) = 0.0225\) So, the probability of both people in the marriage being Rh-negative is \(0.0225\) or \(2.25 \%\). This means that there is a \(2.25 \%\) chance that their children will be Rh-negative.

Key concepts

Complement Rule

The complement rule is a fundamental concept in probability theory. It helps us determine the probability of the occurrence of the opposite event by subtracting the probability of the given event from one.
In the context of the exercise, we started off with the probability of a person having Rh-positive blood, which was given as 85%. To find the probability of a person being Rh-negative, we applied the complement rule:

• First, we acknowledged that the total probability must add up to 100% (or 1 in decimal form).
• Therefore, to find the probability of Rh-negative, we subtracted the probability of being Rh-positive from 1.
• This calculation was done using the formula: \[ P(\text{Rh-negative}) = 1 - P(\text{Rh-positive}) \]
• Substituting the given value: \[ P(\text{Rh-negative}) = 1 - 0.85 \]
• Resulting in: \[ P(\text{Rh-negative}) = 0.15 \]

This means there's a 15% chance of a person having Rh-negative blood. Using the complement rule ensures we don't overlook the total probability, and it provides a simple way to find the likelihood of opposite events.

Independent Events

Independent events in probability imply that the occurrence of one event has no effect on the other. Understanding this concept makes it easier to calculate the combined probability of two such events occurring simultaneously.
In our exercise, the probability of each individual being Rh-negative is independent of the other because one person's blood type does not influence the other's. This property allows us to multiply the individual probabilities to find the joint probability.

• Two events, A and B, are independent if the probability of both happening is equal to the product of their individual probabilities.
• In mathematical terms, if events A and B are independent: \[ P(A \text{ and } B) = P(A) \times P(B) \]
• For our scenario: \[ P(\text{Both Rh-negative}) = P(\text{Rh-negative Person 1}) \times P(\text{Rh-negative Person 2}) \]
• This relationship underscores the independence: \[ P(\text{Both Rh-negative}) = 0.15 \times 0.15 \]
• Concluding with: \[ P(\text{Both Rh-negative}) = 0.0225 \]

Recognizing events as independent is crucial for accurately determining combined probabilities in various real-world scenarios.

Probability Calculation

Calculating probability involves understanding the likelihood of potential outcomes. Importantly, it requires the differentiation between independent and dependent events.
In the case of two individuals both being Rh-negative, the approach revolved around using the already determined individual probabilities. Here's a concise breakdown of how this works:

• Firstly, the probability of one individual being Rh-negative was found using the complement rule: 0.15 for 15%.
• The probability of another independent individual also being Rh-negative is identically 0.15.
• Because these calculations involve independent events, we multiply these probabilities: \[ P(\text{Both Rh-negative}) = 0.15 \times 0.15 \]
• The result of this multiplication, 0.0225, represents a 2.25% probability. This means there's just a slight chance that both individuals in a marriage are Rh-negative, which directly affects the trait in their children.

Overall, probability calculations are foundational in statistics. They allow for predictions and planning based on the likelihood of specific outcomes, an essential capability across diverse applications.