Strawberry plains DMV has a total of 10 written (knowledge) driving test sets and has recently changed its collection of test sets for new drivers. In this regard, now there are 2 high-difficulty test sets, 5 medium difficulty test sets, and 3 low-difficulty test sets. The probability that a new driver passes the most difficult test is 1/9; if it is medium difficulty, 3/5; and if it is low difficulty 3/4. Suppose a new driver is randomly selected. Find the probability that they took one of the most difficult test sets if they passed the test.
The Correct Answer and Explanation is :
To find the probability that a new driver took one of the most difficult test sets given that they passed the test, we use Bayes’ Theorem.
Step 1: Define Events
- Let ( H_1 ) be the event that a driver takes a high-difficulty test.
- Let ( H_2 ) be the event that a driver takes a medium-difficulty test.
- Let ( H_3 ) be the event that a driver takes a low-difficulty test.
- Let ( P ) be the event that the driver passes the test.
We know:
[
P(H_1) = \frac{2}{10}, \quad P(H_2) = \frac{5}{10}, \quad P(H_3) = \frac{3}{10}
]
[
P(P | H_1) = \frac{1}{9}, \quad P(P | H_2) = \frac{3}{5}, \quad P(P | H_3) = \frac{3}{4}
]
Step 2: Find Total Probability of Passing
Using the Law of Total Probability:
[
P(P) = P(P | H_1) P(H_1) + P(P | H_2) P(H_2) + P(P | H_3) P(H_3)
]
Substituting values:
[
P(P) = \left(\frac{1}{9} \times \frac{2}{10} \right) + \left(\frac{3}{5} \times \frac{5}{10} \right) + \left(\frac{3}{4} \times \frac{3}{10} \right)
]
[
= \frac{2}{90} + \frac{15}{50} + \frac{9}{40}
]
Converting to a common denominator (1800):
[
= \frac{40}{1800} + \frac{540}{1800} + \frac{405}{1800} = \frac{985}{1800}
]
Step 3: Find Conditional Probability Using Bayes’ Theorem
[
P(H_1 | P) = \frac{P(P | H_1) P(H_1)}{P(P)}
]
[
= \frac{\frac{2}{90}}{\frac{985}{1800}}
]
[
= \frac{2}{90} \times \frac{1800}{985} = \frac{3600}{88650} = \frac{4}{197}
]
Thus, the probability that a driver took one of the most difficult test sets given that they passed is 4/197.
Explanation:
This problem applies Bayes’ Theorem, which helps calculate conditional probabilities. The intuition is that passing rates differ across difficulty levels. While most tests are of medium difficulty, the low pass rate of difficult tests makes it unlikely that a passing driver took the hardest test. After performing the calculations, we conclude that the probability of having taken a high-difficulty test given a passing result is 4/197, or approximately 0.0203 (2.03%).
Here is the digital illustration of a new driver taking a written driving test at the DMV. Let me know if you need any modifications or additional details!