Max and his sister went snorkeling this summer and took photos of individual fish with their waterproof camera. They printed their favorite 19 photos, 12 of which showed yellow fish and 6 of which showed black fish.

If Max randomly chooses 13 of the photos to hang in his room, what is the probability that 8 of the chosen photos are of yellow fish and 5 are of black fish?

Write your answer as a decimal rounded to four decimal places.

To find the probability that Max randomly chooses 13 photos with exactly 8 of them showing yellow fish and 5 showing black fish, we need to use combinations to calculate the number of ways to choose the desired photos and then divide by the total number of ways to select 13 photos from the 19.

### Step 1: Total ways to select 13 photos from 19
The total number of ways to select 13 photos out of 19 is given by:

\[
\binom{19}{13} = \binom{19}{6}
\]

This simplification is possible because \(\binom{n}{r} = \binom{n}{n-r}\).

### Step 2: Ways to select 8 yellow fish photos
Next, calculate the number of ways to select 8 yellow fish photos out of the 12 available:

\[
\binom{12}{8} = \binom{12}{4}
\]

### Step 3: Ways to select 5 black fish photos
Now, calculate the number of ways to select 5 black fish photos out of the 6 available:

\[
\binom{6}{5} = \binom{6}{1}
\]

### Step 4: Multiply the combinations for yellow and black fish
To find the total number of ways to select 8 yellow fish and 5 black fish photos, multiply the results from Step 2 and Step 3:

\[
\binom{12}{8} \times \binom{6}{5}
\]

### Step 5: Probability calculation
Finally, the probability is given by:

\[
P = \frac{\binom{12}{8} \times \binom{6}{5}}{\binom{19}{13}}
\]

Now, let's calculate these values:The probability that Max randomly chooses 13 photos with exactly 8 showing yellow fish and 5 showing black fish is **0.1095**, rounded to four decimal places.

Would you like more details or have any other questions? Here are some related questions:

1. How do you calculate the probability when selecting photos with different counts?
2. What would be the probability if Max wanted 7 yellow and 6 black fish photos?
3. How does the probability change if there are more photos of black fish?
4. What is the probability of selecting all yellow fish photos?
5. How would the calculation change if Max wanted to hang fewer photos?

**Tip:** When solving probability problems involving selections, use combinations when the order does not matter and consider breaking down the problem into smaller steps for easier calculation.

Learn how to calculate the probability of Max randomly choosing 13 photos with exactly 8 showing yellow fish and 5 showing black fish from a set of 19 photos. Understand the application of combinations and probability in this problem. Suitable for advanced high school students.

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