Let's address each part of the problem step by step.
### 10.2.1 Probability of Choosing "Midnight" from the Drama Category:
Nametso must choose one DVD from the Drama category, which contains the following six DVDs:
1. Last Hero
2. Midnight
3. Stranger Calls
4. Missing in Action
5. Only 40 Seconds Left
There are a total of 6 DVDs in the Drama category.
The probability of choosing any one DVD is given by the formula:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \][/tex]
For choosing "Midnight":
[tex]\[ \text{Probability of choosing Midnight} = \frac{1}{6} \approx 0.1667 \][/tex]
### 10.2.2 Number of Different Selections (One Drama, One Romance, One Comedy):
To find the total number of different selections Nametso can make if she chooses one DVD from each category (Drama, Romance, Comedy), we need to determine the total number of DVDs in each category.
From the table:
- Total number of Drama DVDs = 6
- Total number of Romance DVDs = 5
- Total number of Comedy DVDs = 3
The total number of different selections she can make is calculated by multiplying the number of choices in each category:
[tex]\[ \text{Different selections} = \text{Total number of Drama DVDs} \times \text{Total number of Romance DVDs} \times \text{Total number of Comedy DVDs} \][/tex]
[tex]\[ \text{Different selections} = 6 \times 5 \times 3 = 90 \][/tex]
There are 90 different possible selections.
### 10.2.3 Probability of Selecting "Last Hero" and "Laughing Dragon":
We now need to find the probability that Nametso will have "Last Hero" (from the Drama category) and "Laughing Dragon" (from the Comedy category) as part of her selection.
Firstly, we calculate the probability of choosing each of these DVDs individually:
- Probability of choosing "Last Hero" from the Drama category:
[tex]\[ \text{Probability of choosing Last Hero} = \frac{1}{6} \][/tex]
- Probability of choosing "Laughing Dragon" from the Comedy category:
[tex]\[ \text{Probability of choosing Laughing Dragon} = \frac{1}{3} \][/tex]
Since the selection of a Drama DVD and a Comedy DVD are independent events, the probability of both events happening together is the product of their individual probabilities:
[tex]\[ \text{Probability of both} = \text{Probability of choosing Last Hero} \times \text{Probability of choosing Laughing Dragon} \][/tex]
[tex]\[ \text{Probability of both} = \frac{1}{6} \times \frac{1}{3} = \frac{1}{18} \approx 0.0556 \][/tex]
### Summary of Results:
1. Probability of choosing "Midnight" from Drama: 0.1667 (or [tex]\(\frac{1}{6}\)[/tex])
2. Number of different selections (one Drama, one Romance, one Comedy): 90
3. Probability of selecting "Last Hero" and "Laughing Dragon": 0.0556 (or [tex]\(\frac{1}{18}\)[/tex])
