In a survey of people who recently bought a laptop, 45% said they were looking for a large screen, 31% said they were looking for a fast processor, and 58% said they wanted a large screen or a fast processor. If a survey respondent is selected at random, what is the probability that the respondent wanted both a large screen and a fast processor?
The correct answer and explanation is:
To find the probability that a respondent wanted both a large screen and a fast processor, we use the formula for the union of two sets in probability: P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) – P(A \cap B)
Where:
- P(A)P(A) = probability of wanting a large screen = 45% = 0.45
- P(B)P(B) = probability of wanting a fast processor = 31% = 0.31
- P(A∪B)P(A \cup B) = probability of wanting either a large screen or a fast processor = 58% = 0.58
- P(A∩B)P(A \cap B) = probability of wanting both
We rearrange the formula to solve for P(A∩B)P(A \cap B): P(A∩B)=P(A)+P(B)−P(A∪B)P(A \cap B) = P(A) + P(B) – P(A \cup B)
Substitute the known values: P(A∩B)=0.45+0.31−0.58=0.76−0.58=0.18P(A \cap B) = 0.45 + 0.31 – 0.58 = 0.76 – 0.58 = 0.18
✅ Final Answer:
The probability that a respondent wanted both a large screen and a fast processor is 0.18 or 18%.
📘 Explanation (300 words):
This is a classic example of calculating the intersection of two overlapping groups using the principle of inclusion and exclusion in probability. The goal is to avoid double-counting people who belong to both categories (those who wanted both features).
Here, we are given the individual probabilities of wanting a large screen (45%) and a fast processor (31%), and we are also given the probability that a person wanted either of the two features (58%). The word “or” in probability means we include people who want one or both features.
However, if we just added the 45% and 31% directly, we’d get 76%, which overlaps the people who wanted both features. That’s why we subtract the overlap (intersection) to get the correct total. By rearranging the formula, we can isolate and calculate this intersection.
So, using: P(Large Screen or Fast Processor)=P(Large Screen)+P(Fast Processor)−P(Both)P(\text{Large Screen or Fast Processor}) = P(\text{Large Screen}) + P(\text{Fast Processor}) – P(\text{Both})
and solving for “both,” we get the overlap as 18%.
This type of calculation is useful in analyzing surveys, marketing data, or any scenario where people may have multiple preferences or attributes, and we need to know how many share both.