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There are 13 college students running for student government class president. The candidates include 5 biology majors. If 9 of the candidates are randomly chosen to give the first 9 speeches, what is the probability that exactly 2 of the chosen candidates are biology majors? Write your answer as a decimal rounded to four decimal places. ◻

There are 1313 college students running for student government class president. The candidates include 55 biology majors.\newlineIf 99 of the candidates are randomly chosen to give the first 99 speeches, what is the probability that exactly 22 of the chosen candidates are biology majors?\newlineWrite your answer as a decimal rounded to four decimal places.\newline□ \square

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Q. There are 1313 college students running for student government class president. The candidates include 55 biology majors.\newlineIf 99 of the candidates are randomly chosen to give the first 99 speeches, what is the probability that exactly 22 of the chosen candidates are biology majors?\newlineWrite your answer as a decimal rounded to four decimal places.\newline□ \square
  1. Identify Total Number: Identify the total number of ways to choose 99 candidates out of 1313 without regard to the major.\newlineWe use the combination formula C(n,k)=n!k!(n−k)!C(n, k) = \frac{n!}{k!(n-k)!} where nn is the total number of items, kk is the number of items to choose, and “!” denotes factorial.\newlineTotal number of ways to choose 99 candidates out of 1313 is C(13,9)C(13, 9).\newlineCalculate C(13,9)=13!9!(13−9)!=13!9!4!=13×12×11×104×3×2×1C(13, 9) = \frac{13!}{9!(13-9)!} = \frac{13!}{9!4!} = \frac{13\times12\times11\times10}{4\times3\times2\times1}.
  2. Calculate Combination: Identify the number of ways to choose 22 biology majors out of 55. We use the same combination formula C(n,k)C(n, k). Number of ways to choose 22 biology majors out of 55 is C(5,2)C(5, 2). Calculate C(5,2)=5!2!(5−2)!=5!2!3!=(5×4)(2×1)C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{(5\times4)}{(2\times1)}.
  3. Identify Number Biology Majors: Identify the number of ways to choose the remaining 77 candidates from the non-biology majors.\newlineThere are 13−5=813 - 5 = 8 non-biology majors.\newlineNumber of ways to choose 77 candidates out of 88 non-biology majors is C(8,7)C(8, 7).\newlineCalculate C(8,7)=8!(7!(8−7)!)=8!(7!1!)=81.C(8, 7) = \frac{8!}{(7!(8-7)!)} = \frac{8!}{(7!1!)} = \frac{8}{1}.
  4. Calculate Number Non-Biology Majors: Calculate the number of ways to choose exactly 22 biology majors and 77 non-biology majors.\newlineMultiply the results from Step 22 and Step 33.\newlineNumber of favorable outcomes = C(5,2)×C(8,7)C(5, 2) \times C(8, 7).\newlineCalculate the number of favorable outcomes = (5×4)/(2×1)×8/1=10×8=80(5\times4) / (2\times1) \times 8 / 1 = 10 \times 8 = 80.
  5. Calculate Number Favorable Outcomes: Calculate the probability that exactly 22 of the chosen candidates are biology majors.\newlineProbability =Number of favorable outcomesTotal number of ways to choose 9 candidates= \frac{\text{Number of favorable outcomes}}{\text{Total number of ways to choose } 9 \text{ candidates}}.\newlineCalculate the probability =80C(13,9)= \frac{80}{C(13, 9)}.\newlineFrom Step 11, we have C(13,9)=(13×12×11×10)(4×3×2×1)C(13, 9) = \frac{(13\times12\times11\times10)}{(4\times3\times2\times1)}.\newlineCalculate the probability =80((13×12×11×10)(4×3×2×1))= \frac{80}{\left(\frac{(13\times12\times11\times10)}{(4\times3\times2\times1)}\right)}.
  6. Calculate Probability: Simplify the probability and round to four decimal places.\newlineCalculate the probability = 80(13×12×11×104×3×2×1)=8013×12×11×10×4×3×2×11\frac{80}{\left(\frac{13\times12\times11\times10}{4\times3\times2\times1}\right)} = \frac{80}{13\times12\times11\times10} \times \frac{4\times3\times2\times1}{1}.\newlineSimplify the probability = 8017160=0.004662004662004662\frac{80}{17160} = 0.004662004662004662.\newlineRound to four decimal places = 0.00470.0047.

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