Question 1176351: During a 52-week period, a company paid overtime wages for 18 weeks and hired temporary help for 9 weeks. During 5 weeks, the company paid overtime and hired temporary help. Let: O = overtime, T = temporary help.
a. What is the probability P(O OR T) ?
Round your answer to the nearest hundredth.
b. Are O and T independent?
c. Are O and T mutually exclusive?
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step.
**Given Information:**
* Total weeks: 52
* Overtime (O) weeks: 18
* Temporary help (T) weeks: 9
* Overtime and temporary help (O ∩ T) weeks: 5
**a) What is the probability P(O OR T)?**
We'll use the formula for the probability of the union of two events:
* P(O ∪ T) = P(O) + P(T) - P(O ∩ T)
1. **Calculate P(O):**
* P(O) = Number of overtime weeks / Total weeks
* P(O) = 18 / 52
2. **Calculate P(T):**
* P(T) = Number of temporary help weeks / Total weeks
* P(T) = 9 / 52
3. **Calculate P(O ∩ T):**
* P(O ∩ T) = Number of weeks with both / Total weeks
* P(O ∩ T) = 5 / 52
4. **Calculate P(O ∪ T):**
* P(O ∪ T) = (18 / 52) + (9 / 52) - (5 / 52)
* P(O ∪ T) = (18 + 9 - 5) / 52
* P(O ∪ T) = 22 / 52
* P(O ∪ T) = 11 / 26
* P(O ∪ T) ≈ 0.4230769
5. **Round to the nearest hundredth:**
* P(O ∪ T) ≈ 0.42
**b) Are O and T independent?**
Two events are independent if P(O ∩ T) = P(O) * P(T).
1. **Calculate P(O) * P(T):**
* P(O) * P(T) = (18 / 52) * (9 / 52)
* P(O) * P(T) = 162 / 2704
* P(O) * P(T) ≈ 0.0599
2. **Compare P(O ∩ T) and P(O) * P(T):**
* P(O ∩ T) = 5 / 52 ≈ 0.0961
* P(O) * P(T) ≈ 0.0599
3. **Conclusion:**
* Since P(O ∩ T) ≠ P(O) * P(T), the events O and T are **not independent**.
**c) Are O and T mutually exclusive?**
Two events are mutually exclusive if P(O ∩ T) = 0.
1. **Check P(O ∩ T):**
* P(O ∩ T) = 5 / 52
2. **Conclusion:**
* Since P(O ∩ T) ≠ 0, the events O and T are **not mutually exclusive**.
**Answers:**
* **a) P(O OR T) ≈ 0.42**
* **b) No, O and T are not independent.**
* **c) No, O and T are not mutually exclusive.**
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
As the problem is worded, printed and presented in the post, it makes no sense mathematically.
In other words, it is soup of words with no sense.
The " solution " by @CPhill in his post is nonsense raised in degree 2.
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