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 solve this problem, we can use the principle of inclusion-exclusion for probabilities.
Definitions:
- P(A)P(A): Probability of wanting a large screen = 0.450.45
- P(B)P(B): Probability of wanting a fast processor = 0.310.31
- P(A∪B)P(A \cup B): Probability of wanting a large screen or a fast processor = 0.580.58
- P(A∩B)P(A \cap B): Probability of wanting both a large screen and a fast processor (to be calculated).
The formula for the union of two events is: P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) – P(A \cap B)
Rearranging to find 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)
Calculation:
Substitute the given values: P(A∩B)=0.45+0.31−0.58P(A \cap B) = 0.45 + 0.31 – 0.58 P(A∩B)=0.18P(A \cap B) = 0.18
Final Answer:
The probability that a respondent wanted both a large screen and a fast processor is 0.18 (or 18%).
Explanation:
This problem applies the principle of inclusion-exclusion to determine the overlap between two groups: people who desire a large screen and those who desire a fast processor. While P(A)P(A) and P(B)P(B) separately give the probabilities for each preference, their union P(A∪B)P(A \cup B) includes all respondents who desired either one or both. However, counting P(A)+P(B)P(A) + P(B) without adjustment double-counts the individuals who fall into both categories.
The subtraction of P(A∪B)P(A \cup B) corrects this overcounting and isolates the shared proportion of the population that wanted both features. Hence, the result 0.180.18 reflects the probability of a respondent desiring both a large screen and a fast processor. This approach ensures that all possible outcomes are accounted for accurately and avoids double-counting overlapping preferences.