Police plan to enforce speed limits by using radar traps at four different locations within the city limits. The radar traps at each of the locations L1, L2, L3, and L4 will be operated 40%, 30%, 20%, and 30% of the time. If a person who is speeding on her way to work has probabilities of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations, what is the probability that she will receive a speeding ticket?

We need to calculate the probability that a person speeding will receive a ticket. The person passes through certain locations with known probabilities, and the radar traps are operated at those locations with other probabilities. The problem can be solved using the **law of total probability**, which states:

\[
P(\text{Ticket}) = P(\text{Ticket at L1}) + P(\text{Ticket at L2}) + P(\text{Ticket at L3}) + P(\text{Ticket at L4})
\]

Where each term represents the joint probability of the person passing through a location and getting a ticket at that location.

### Step 1: Probabilities of being caught at each location

The probability that the person receives a ticket at a specific location is the product of two probabilities:
- The probability the person passes through the location.
- The probability the radar is in operation at that location.

So, the total probability is:

\[
P(\text{Ticket}) = P(\text{L1}) \times P(\text{Radar at L1}) + P(\text{L2}) \times P(\text{Radar at L2}) + P(\text{L3}) \times P(\text{Radar at L3}) + P(\text{L4}) \times P(\text{Radar at L4})
\]

### Step 2: Insert the known values

- \( P(\text{Radar at L1}) = 0.40 \), \( P(\text{L1}) = 0.2 \)
- \( P(\text{Radar at L2}) = 0.30 \), \( P(\text{L2}) = 0.1 \)
- \( P(\text{Radar at L3}) = 0.20 \), \( P(\text{L3}) = 0.5 \)
- \( P(\text{Radar at L4}) = 0.30 \), \( P(\text{L4}) = 0.2 \)

Now calculate each term:

\[
P(\text{Ticket at L1}) = 0.2 \times 0.4 = 0.08
\]
\[
P(\text{Ticket at L2}) = 0.1 \times 0.3 = 0.03
\]
\[
P(\text{Ticket at L3}) = 0.5 \times 0.2 = 0.10
\]
\[
P(\text{Ticket at L4}) = 0.2 \times 0.3 = 0.06
\]

### Step 3: Sum the probabilities

\[
P(\text{Ticket}) = 0.08 + 0.03 + 0.10 + 0.06 = 0.27
\]

### Final Answer:
The probability that the person will receive a speeding ticket is **0.27**, or 27%.

---

Would you like more details or clarifications? Here are some related questions to explore:

1. How would the answer change if the probabilities of passing through each location were different?
2. What if the radar operation times were equally distributed at all locations?
3. How could this problem be modeled using conditional probability?
4. What is the probability the person passes through two locations and gets caught at both?
5. Can this problem be extended to more than four locations?

**Tip:** The law of total probability is a powerful tool for dealing with problems involving multiple possible events, especially when you know the probabilities of those events occurring!

Determine the probability of receiving a speeding ticket when passing through multiple radar locations in the city. This problem utilizes the Law of Total Probability, covering key concepts in probability theory. Suitable for students in Grades 9-12.

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