Math Problem Statement

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.

Solution

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:

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

where n=30n = 30 and r=4r = 4:

(304)=30×29×28×274×3×2×1\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:

(222)=22×212×1\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:

(72)=7×62×1\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:

(222)×(72)\binom{22}{2} \times \binom{7}{2}

Step 4: Probability calculation

Finally, the probability is given by:

P=(222)×(72)(304)P = \frac{\binom{22}{2} \times \binom{7}{2}}{\binom{30}{4}}

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

  1. (304)\binom{30}{4}
  2. (222)\binom{22}{2}
  3. (72)\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.

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Math Problem Analysis

Mathematical Concepts

Probability
Combinatorics

Formulas

Combinations formula

Theorems

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Suitable Grade Level

High School