An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probabilities of an accident involving a scooter driver car driver and a truck driver are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. The probability that the person is a scooter driver is
This problem involves calculating a conditional probability using Bayes' theorem. We are given the number of drivers in different categories (scooter, car, truck) and the probability of an accident for each category. We need to find the probability that a person involved in an accident is a scooter driver.
Understanding the Problem and Defining Events
Let's define the events involved in this problem:
- Let \(S\) be the event that the insured person is a scooter driver.
- Let \(C\) be the event that the insured person is a car driver.
- Let \(T\) be the event that the insured person is a truck driver.
- Let \(A\) be the event that the insured person meets with an accident.
Calculating Prior Probabilities
First, we need to find the total number of insured persons and the probability of selecting a person from each category (these are the prior probabilities).
Total number of insured persons = Number of scooter drivers + Number of car drivers + Number of truck drivers
Total insured persons = 2000 + 4000 + 6000 = 12000
Now, we can calculate the prior probability for each category:
- Probability of a person being a scooter driver, \(P(S) = \frac{\text{Number of scooter drivers}}{\text{Total insured persons}} = \frac{2000}{12000} = \frac{1}{6}\)
- Probability of a person being a car driver, \(P(C) = \frac{\text{Number of car drivers}}{\text{Total insured persons}} = \frac{4000}{12000} = \frac{4}{12} = \frac{1}{3}\)
- Probability of a person being a truck driver, \(P(T) = \frac{\text{Number of truck drivers}}{\text{Total insured persons}} = \frac{6000}{12000} = \frac{6}{12} = \frac{1}{2}\)
Identifying Conditional Probabilities of Accident
The problem gives us the probability of an accident occurring given the type of driver. These are the conditional probabilities:
- Probability of an accident given the person is a scooter driver, \(P(A|S) = 0.01 = \frac{1}{100}\)
- Probability of an accident given the person is a car driver, \(P(A|C) = 0.03 = \frac{3}{100}\)
- Probability of an accident given the person is a truck driver, \(P(A|T) = 0.15 = \frac{15}{100}\)
Calculating the Total Probability of an Accident
To use Bayes' theorem, we first need to find the total probability of an accident occurring, \(P(A)\). We can find this using the Law of Total Probability:
\(P(A) = P(A|S)P(S) + P(A|C)P(C) + P(A|T)P(T)\)
Substitute the values we calculated:
\(P(A) = \left(\frac{1}{100}\right) \times \left(\frac{1}{6}\right) + \left(\frac{3}{100}\right) \times \left(\frac{1}{3}\right) + \left(\frac{15}{100}\right) \times \left(\frac{1}{2}\right)\)
\(P(A) = \frac{1}{600} + \frac{3}{300} + \frac{15}{200}\)
Find a common denominator, which is 600:
\(P(A) = \frac{1}{600} + \frac{3 \times 2}{300 \times 2} + \frac{15 \times 3}{200 \times 3}\)
\(P(A) = \frac{1}{600} + \frac{6}{600} + \frac{45}{600}\)
\(P(A) = \frac{1 + 6 + 45}{600} = \frac{52}{600}\)
Applying Bayes' Theorem
We want to find the probability that the person involved in the accident is a scooter driver, i.e., \(P(S|A)\). According to Bayes' theorem, this is given by:
\(P(S|A) = \frac{P(A|S)P(S)}{P(A)}\)
Substitute the calculated values into the formula:
\(P(S|A) = \frac{\left(\frac{1}{100}\right) \times \left(\frac{1}{6}\right)}{\frac{52}{600}}\)
\(P(S|A) = \frac{\frac{1}{600}}{\frac{52}{600}}\)
\(P(S|A) = \frac{1}{600} \times \frac{600}{52}\)
\(P(S|A) = \frac{1}{52}\)
Thus, the probability that the person is a scooter driver given that they met with an accident is \(\frac{1}{52}\).
Summary of Probabilities
| Driver Type | Number of Drivers | Prior Probability P(Driver) | Accident Probability P(Accident|Driver) | Joint Probability P(Accident and Driver) = P(A|Driver)P(Driver) |
|---|---|---|---|---|
| Scooter (S) | 2000 | \(P(S) = \frac{2000}{12000} = \frac{1}{6}\) | \(P(A|S) = 0.01 = \frac{1}{100}\) | \(P(A \text{ and } S) = \frac{1}{100} \times \frac{1}{6} = \frac{1}{600}\) |
| Car (C) | 4000 | \(P(C) = \frac{4000}{12000} = \frac{1}{3}\) | \(P(A|C) = 0.03 = \frac{3}{100}\) | \(P(A \text{ and } C) = \frac{3}{100} \times \frac{1}{3} = \frac{3}{300} = \frac{6}{600}\) |
| Truck (T) | 6000 | \(P(T) = \frac{6000}{12000} = \frac{1}{2}\) | \(P(A|T) = 0.15 = \frac{15}{100}\) | \(P(A \text{ and } T) = \frac{15}{100} \times \frac{1}{2} = \frac{15}{200} = \frac{45}{600}\) |
Total probability of accident \(P(A) = P(A \text{ and } S) + P(A \text{ and } C) + P(A \text{ and } T) = \frac{1}{600} + \frac{6}{600} + \frac{45}{600} = \frac{52}{600}\).
Using Bayes' Theorem for \(P(S|A)\):
\(P(S|A) = \frac{P(A \text{ and } S)}{P(A)} = \frac{\frac{1}{600}}{\frac{52}{600}} = \frac{1}{52}\)
Final Answer
The probability that the person involved in the accident is a scooter driver is \(\frac{1}{52}\).
Revision Table: Accident Probability Calculation
| Step | Description | Formula/Calculation |
|---|---|---|
| 1 | Calculate total insured drivers. | \(2000 + 4000 + 6000 = 12000\) |
| 2 | Calculate prior probabilities P(S), P(C), P(T). | \(P(S) = \frac{2000}{12000} = \frac{1}{6}\), \(P(C) = \frac{4000}{12000} = \frac{1}{3}\), \(P(T) = \frac{6000}{12000} = \frac{1}{2}\) |
| 3 | Identify conditional probabilities P(A|S), P(A|C), P(A|T). | \(P(A|S) = 0.01\), \(P(A|C) = 0.03\), \(P(A|T) = 0.15\) |
| 4 | Calculate total probability of accident P(A). | \(P(A) = P(A|S)P(S) + P(A|C)P(C) + P(A|T)P(T)\) \(P(A) = (0.01)(\frac{1}{6}) + (0.03)(\frac{1}{3}) + (0.15)(\frac{1}{2})\) \(P(A) = \frac{1}{600} + \frac{3}{300} + \frac{15}{200} = \frac{1+6+45}{600} = \frac{52}{600}\) |
| 5 | Apply Bayes' Theorem for P(S|A). | \(P(S|A) = \frac{P(A|S)P(S)}{P(A)} = \frac{(0.01)(\frac{1}{6})}{\frac{52}{600}} = \frac{\frac{1}{600}}{\frac{52}{600}} = \frac{1}{52}\) |
Additional Information: Bayes' Theorem and Conditional Probability
This problem is a classic example where Bayes' theorem is used. Bayes' theorem allows us to update the probability of a hypothesis (in this case, the type of driver) based on new evidence (the occurrence of an accident).
Conditional Probability: \(P(X|Y)\) is the probability of event X occurring given that event Y has already occurred. In this problem, \(P(A|S)\) is the probability of an accident given the driver is a scooter driver.
Law of Total Probability: If \(E_1, E_2, \dots, E_n\) are mutually exclusive and exhaustive events (they cover all possibilities and don't overlap), then the probability of any event A can be calculated as \(P(A) = \sum_{i=1}^n P(A|E_i)P(E_i)\). In our case, the events S, C, and T are mutually exclusive and exhaustive categories of drivers.
Bayes' Theorem: It states that \(P(E_i|A) = \frac{P(A|E_i)P(E_i)}{P(A)}\). This formula helps us find the probability of a specific event \(E_i\) occurring given that event A has occurred. In this problem, we used it to find \(P(S|A)\).
Bayes' theorem is widely used in various fields, including medical diagnosis, spam filtering, and machine learning.
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