Problem 67
Question
A class consists of 80 students, 25 of them are girls. If 10 of the students are rich and 20 of the students are fair complexioned, then the probability of selecting a fair complexioned rich girl from the class (assuming three traits as independent) is (A) \(1 / 10\) (B) \(1 / 32\) (C) \(5 / 512\) (D) \(7 / 512\)
Step-by-Step Solution
Verified Answer
Option (C) \( \frac{5}{512} \).
1Step 1: Understand Total Opportunities
First, identify the total number of opportunities for selecting any student from the class. The total number of students in the class is 80.
2Step 2: Identify Probability for Each Trait
Since the traits are independent, determine the individual probability of the desired traits: - Probability of selecting a girl = \( \frac{25}{80} \)- Probability of selecting a rich student = \( \frac{10}{80} \)- Probability of selecting a fair complexioned student = \( \frac{20}{80} \)
3Step 3: Calculate Combined Probability
As the traits are independent, multiply the individual probabilities to find the combined probability of selecting a fair complexioned rich girl:\[P(\text{Girl} \cap \text{Rich} \cap \text{Fair Complexioned}) = \frac{25}{80} \times \frac{10}{80} \times \frac{20}{80}\]This simplifies to:\[\frac{25}{80} \times \frac{10}{80} \times \frac{20}{80} = \frac{5000}{512000}= \frac{5}{512}\]
4Step 4: Choose the Correct Answer
Compare the calculated probability with the given options. The calculated probability is \( \frac{5}{512} \), which matches option (C).
Key Concepts
Independent EventsCombined ProbabilityConditional Probability
Independent Events
In probability, when we say that events are independent, it means that the occurrence of one event does not affect the occurrence of another. For example, in the context of our class selection problem, the fact that a student is a girl does not impact the likelihood of her being rich or fair complexioned. Independent events have no influence on each other, and their probabilities can be calculated separately.
Independent events are crucial when calculating combined probabilities. Suppose you flip two coins; the outcome of one coin does not influence the outcome of the other. The probability of getting heads on one coin is independent of getting heads on the other. Hence, the combined probability of both events occurring can be found by simply multiplying their individual probabilities.
For this reason, if we can confirm that events in a scenario are independent, it simplifies the process of calculating complex probabilities, as we do not need to consider how one event might alter another.
Independent events are crucial when calculating combined probabilities. Suppose you flip two coins; the outcome of one coin does not influence the outcome of the other. The probability of getting heads on one coin is independent of getting heads on the other. Hence, the combined probability of both events occurring can be found by simply multiplying their individual probabilities.
For this reason, if we can confirm that events in a scenario are independent, it simplifies the process of calculating complex probabilities, as we do not need to consider how one event might alter another.
Combined Probability
Combined probability refers to the probability of two or more independent events happening at the same time. As seen in the original exercise, to determine the probability of multiple independent traits or events occurring simultaneously, we multiply their separate probabilities.
For instance, in the class exercise, we were looking for the probability that a student is both a girl, rich, and fair complexioned. Given the independence of these traits, we can use the formula for combined probability:
\[ P(\text{Girl} \cap \text{Rich} \cap \text{Fair Complexioned}) = \frac{25}{80} \times \frac{10}{80} \times \frac{20}{80} \]
The result is \(\frac{5}{512}\), reflecting the probability of all traits being present in a student chosen randomly from the class.
For instance, in the class exercise, we were looking for the probability that a student is both a girl, rich, and fair complexioned. Given the independence of these traits, we can use the formula for combined probability:
- Probability of a girl: \(\frac{25}{80}\)
- Probability of being rich: \(\frac{10}{80}\)
- Probability of fair complexion: \(\frac{20}{80}\)
\[ P(\text{Girl} \cap \text{Rich} \cap \text{Fair Complexioned}) = \frac{25}{80} \times \frac{10}{80} \times \frac{20}{80} \]
The result is \(\frac{5}{512}\), reflecting the probability of all traits being present in a student chosen randomly from the class.
Conditional Probability
Conditional probability deals with the chances of an event occurring given that another event has already occurred. It is essential when dealing with events that are not independent. However, in our exercise, the problem specified that the events were independent, so conditional probability did not directly apply.
Still, understanding conditional probability is key when approaching more complex systems. For example, if the exercise had considered dependent traits, such as rich students being more likely to be fair complexioned, conditional probability would become relevant. In such cases, the probability of selecting a fair complexioned student who is rich might depend upon this pre-existing condition – that they are rich.
Using conditional probability, one would adjust calculations by taking into account how one trait (event) influences the other. The general formula used is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Where \( P(A|B) \) represents the probability of event A occurring given event B has occurred. Mastery of conditional probability will broaden your toolkit for solving probability problems across a variety of scenarios.
Still, understanding conditional probability is key when approaching more complex systems. For example, if the exercise had considered dependent traits, such as rich students being more likely to be fair complexioned, conditional probability would become relevant. In such cases, the probability of selecting a fair complexioned student who is rich might depend upon this pre-existing condition – that they are rich.
Using conditional probability, one would adjust calculations by taking into account how one trait (event) influences the other. The general formula used is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Where \( P(A|B) \) represents the probability of event A occurring given event B has occurred. Mastery of conditional probability will broaden your toolkit for solving probability problems across a variety of scenarios.
Other exercises in this chapter
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