Question

You solved this problem in another question: A video rental store keeps a list of their top 15 movie rentals each week. This week the list includes 6 action, 4 comedies, 3 dramas, and 2 mysteries. The store manager removes a copy of each of the 15 movies from the shelf, then randomly selects 3 of the 15 to show on the display monitors in the store. What is the probability that she selected 2 comedies and 1 action movie? EXPLAIN how the probability would change if two more comedy movies were added to the list in place of the 2 mystery movies.

Answer 1

To find the probability of selecting 2 comedies and 1 action movie from the top 15 rentals, we can use combinations.

### Step 1: Define the Total Numbers
- Total movies: 15 (6 action, 4 comedies, 3 dramas, 2 mysteries)
- We need to choose 3 movies from these 15.

### Step 2: Calculate Total Combinations
The total number of ways to choose 3 movies from 15 is given by the combination formula:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
Thus, the total combinations of choosing 3 from 15 is:
\[
\binom{15}{3} = \frac{15!}{3! \cdot (15-3)!} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = 455
\]

### Step 3: Calculate the Favorable Outcomes
Next, we find the number of ways to select 2 comedies and 1 action movie:
1. **Selecting 2 Comedies**: The number of ways to choose 2 out of 4 comedies is:
\[
\binom{4}{2} = \frac{4!}{2! \cdot (4-2)!} = \frac{4 \times 3}{2 \times 1} = 6
\]

2. **Selecting 1 Action Movie**: The number of ways to choose 1 out of 6 action movies is:
\[
\binom{6}{1} = 6
\]

### Step 4: Calculate Favorable Outcomes
Now we find the total number of favorable outcomes (2 comedies and 1 action):
\[
\text{Favorable Outcomes} = \binom{4}{2} \cdot \binom{6}{1} = 6 \cdot 6 = 36
\]

### Step 5: Calculate the Probability
Now we can calculate the probability:
\[
P(\text{2 comedies and 1 action}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{36}{455}
\]

### Step 6: If Two More Comedy Movies are Added
If we replace the 2 mystery movies with 2 additional comedies, the new counts will be:
- Comedies: 6 (4 original + 2 added)
- Total movies: 15 (still the same)

Now, we need to recalculate the combinations with the updated number of comedies:
1. **Selecting 2 Comedies**: Now the number of ways to choose 2 out of 6 comedies is:
\[
\binom{6}{2} = \frac{6!}{2! \cdot (6-2)!} = \frac{6 \times 5}{2 \times 1} = 15
\]

2. **Selecting 1 Action Movie**: The number of ways to choose 1 out of 6 action movies remains the same:
\[
\binom{6}{1} = 6
\]

3. **Calculate New Favorable Outcomes**:
\[
\text{New Favorable Outcomes} = \binom{6}{2} \cdot \binom{6}{1} = 15 \cdot 6 = 90
\]

### Step 7: Calculate New Probability
Again, the total combinations of choosing 3 from 15 remains 455. So, the new probability is:
\[
P(\text{2 comedies and 1 action with 2 more comedies}) = \frac{90}{455}
\]

### Summary
- Original probability of selecting 2 comedies and 1 action movie: \(\frac{36}{455}\)
- New probability with 2 more comedies added: \(\frac{90}{455}\)

The addition of 2 more comedies increases the probability of selecting 2 comedies and 1 action movie.