Krysta brought 30 bagels to school, 22 of which are poppy seed and 7 of which are blueberry.

If Krysta randomly gives away 4 bagels to teachers, what is the probability that 2 of the chosen bagels are poppy seed and 2 are blueberry?

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

To find the probability that Krysta randomly gives away 4 bagels with exactly 2 being poppy seed and 2 being blueberry, we'll use combinations to determine the number of ways to select the desired bagels and then divide by the total number of ways to choose 4 bagels from the 30.

### Step 1: Total ways to select 4 bagels from 30
The total number of ways to select 4 bagels out of 30 is given by the combination formula:

\[
\binom{n}{r} = \frac{n!}{r!(n-r)!}
\]

where \(n = 30\) and \(r = 4\):

\[
\binom{30}{4} = \frac{30 \times 29 \times 28 \times 27}{4 \times 3 \times 2 \times 1}
\]

### Step 2: Ways to select 2 poppy seed and 2 blueberry bagels
Next, we find the number of ways to select 2 poppy seed bagels out of the 22 available:

\[
\binom{22}{2} = \frac{22 \times 21}{2 \times 1}
\]

And the number of ways to select 2 blueberry bagels out of the 7 available:

\[
\binom{7}{2} = \frac{7 \times 6}{2 \times 1}
\]

### Step 3: Multiply the combinations for poppy seed and blueberry
To find the total number of ways to select 2 poppy seed and 2 blueberry bagels, we multiply the results from Step 2:

\[
\binom{22}{2} \times \binom{7}{2}
\]

### Step 4: Probability calculation
Finally, the probability is given by:

\[
P = \frac{\binom{22}{2} \times \binom{7}{2}}{\binom{30}{4}}
\]

Now, let's calculate these values step by step: 

1. \(\binom{30}{4}\)  
2. \(\binom{22}{2}\)  
3. \(\binom{7}{2}\)  
4. Multiply the combinations for poppy seed and blueberry  
5. Compute the probability. 

Let's compute this:The probability that Krysta randomly gives away 4 bagels with exactly 2 being poppy seed and 2 being blueberry is **0.1770**, rounded to four decimal places.

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

1. How do you calculate combinations in probability problems?
2. What is the probability of selecting all 4 bagels as poppy seed?
3. How does the probability change if the numbers of bagels change?
4. What is the probability of selecting exactly 3 poppy seed bagels?
5. How would the probability change if Krysta gave away 5 bagels instead of 4?

**Tip:** When calculating probabilities involving selections, remember to use combinations (not permutations) as the order of selection does not matter in these cases.

Learn how to calculate the probability of distributing Krysta's bagels, where she randomly gives away 4 bagels with exactly 2 poppy seed and 2 blueberry. This problem involves combinatorics and probability calculations. Suitable for high school students.

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