10. Mariha is evaluating three different driving schools. She asked 50 people who attended one of the schools whether they passed the driving test on the first try. The results of the survey are shown in the table below. P is the probability of randomly selecting a student who passed on the first try from the students who attended A-1 Driving School. Q is the probability of randomly selecting a student who attended A-1 Driving School from the students who passed the driving test on the first try. Which of the following must be true?
Answer: C
P is greater than Q.
In this scenario, P represents the probability of selecting a student who passed the driving test on their first try from all students who attended A-1 Driving School, while Q represents the probability of selecting a student who attended A-1 Driving School from all students who passed the test. Given that P focuses on the subset of students who passed, it inherently has a higher likelihood than Q, which encompasses a broader group.
A) P=Q
This option suggests that the probabilities of P and Q are equal. However, this is not true in this context because P measures a specific outcome (passing the test) among A-1's students, while Q measures a general group (all who passed, which includes students from other schools). Thus, P cannot equal Q.
B) P
This option indicates that the probability P is less than Q. This is incorrect since P, being the probability of selecting a successful student from A-1, should be higher than Q, which includes students from all schools who passed. Therefore, it cannot be true that P is less than Q.
C) P>Q
This statement is correct. P reflects the probability of choosing a successful student from A-1 Driving School, which is likely higher than Q, the probability of choosing any student from the broader category of those who passed. Since students who passed the test include those from other schools, P must be greater than Q, confirming this option.
D) The probabilities cannot be compared.
This option suggests that P and Q are incomparable, which is misleading. While they represent different populations, it is still valid to compare them based on their definitions. In this case, since we know P is derived from a specific group (A-1) and Q from a more general group, they can indeed be compared, making this statement incorrect.
Conclusion
P is definitively greater than Q because P is a targeted measure of success within A-1 Driving School, while Q encompasses a broader student population that includes successful students from various schools. Options A, B, and D fail to recognize the inherent differences between the two probabilities, thus affirming that C is the only correct choice.