There are two traffic lights on Darlene's route from home to work. Let E denote the event that Darlene must stop at the first light, and define the event F in a similar manner for the second light. Suppose that 
P(E) = 0.2,
 
P(F) = 0.2,
 and 
P(E ∩ F) = 0.12.
(a)
What is the probability that Darlene must stop at at least one light; that is, what is the probability of the event 
P(E ∪ F)?
(b)
What is the probability that Darlene doesn't have to stop at either light?
(c)
What is the probability that Darlene must stop at exactly one of the two lights?
(d)
What is the probability that Darlene must stop just at the first light? (Hint: How is the probability of this event related to 
P(E)
 and 
P(E ∩ F)?
 A Venn diagram might help.)


Let's break down the problem and calculate each part using the given probabilities.

Given:
- \( P(E) = 0.2 \) (probability that Darlene must stop at the first light),
- \( P(F) = 0.2 \) (probability that Darlene must stop at the second light),
- \( P(E \cap F) = 0.12 \) (probability that Darlene must stop at both lights).

We are required to calculate the following:

### (a) Probability that Darlene must stop at at least one light, i.e., \( P(E \cup F) \).

Using the inclusion-exclusion formula for the union of two events:
\[
P(E \cup F) = P(E) + P(F) - P(E \cap F)
\]
Substitute the values:
\[
P(E \cup F) = 0.2 + 0.2 - 0.12 = 0.28
\]
So, the probability that Darlene must stop at at least one light is \( \mathbf{0.28} \).

---

### (b) Probability that Darlene doesn't have to stop at either light.

This is the complement of the event that Darlene has to stop at at least one light:
\[
P(\text{No stop}) = 1 - P(E \cup F)
\]
Substitute the value of \( P(E \cup F) \) from part (a):
\[
P(\text{No stop}) = 1 - 0.28 = 0.72
\]
So, the probability that Darlene doesn't have to stop at either light is \( \mathbf{0.72} \).

---

### (c) Probability that Darlene must stop at exactly one of the two lights.

This event occurs when she stops at the first light but not the second, or stops at the second light but not the first. So, we can calculate this as:
\[
P(\text{exactly one stop}) = P(E \cap F^c) + P(E^c \cap F)
\]
Using the following relationships:
- \( P(E \cap F^c) = P(E) - P(E \cap F) \),
- \( P(E^c \cap F) = P(F) - P(E \cap F) \),

Substitute the known values:
\[
P(E \cap F^c) = 0.2 - 0.12 = 0.08,
\]
\[
P(E^c \cap F) = 0.2 - 0.12 = 0.08.
\]
So, the probability of stopping at exactly one light is:
\[
P(\text{exactly one stop}) = 0.08 + 0.08 = 0.16
\]
Thus, the probability that Darlene must stop at exactly one of the two lights is \( \mathbf{0.16} \).

---

### (d) Probability that Darlene must stop just at the first light.

This event is \( P(E \cap F^c) \), which we already calculated in part (c):
\[
P(E \cap F^c) = P(E) - P(E \cap F) = 0.2 - 0.12 = 0.08
\]
Thus, the probability that Darlene must stop just at the first light is \( \mathbf{0.08} \).

---

### Summary of answers:
(a) \( P(E \cup F) = 0.28 \),
(b) \( P(\text{No stop}) = 0.72 \),
(c) \( P(\text{exactly one stop}) = 0.16 \),
(d) \( P(E \cap F^c) = 0.08 \).

Would you like a detailed breakdown of any specific step? Here are some related questions:

1. What if the probability of stopping at each light was different? How would that affect the calculations?
2. How can we use conditional probability to analyze these events further?
3. How would these probabilities change if the two events were independent?
4. Can you explain the inclusion-exclusion principle with more examples?
5. How would a Venn diagram help visualize these probabilities?

**Tip**: For problems involving unions of events, the inclusion-exclusion principle is crucial to avoid double-counting overlaps.

There are two traffic lights on Darlene's route from home to work. Let E denote the event that Darlene must stop at the first light, and define the event F in a similar manner for the second light. Suppose that P(E) = 0.2, P(F) = 0.2, and P(E ∩ F) = 0.12. (a) What is the probability that Darlene must stop at at least one light; that is, what is the probability of the event P(E ∪ F)? (b) What is the probability that Darlene doesn't have to stop at either light? (c) What is the probability that Darlene must stop at exactly one of the two lights? (d) What is the probability that Darlene must stop just at the first light?

Learn how to calculate the probability of stopping at traffic lights on a route using the inclusion-exclusion principle. This problem covers key probability concepts such as event union, intersection, complementary probability, and specific event scenarios. Suitable for students in Grades 10-12.

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