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Last year, Ashley came in 22nd place in her school's Spelling Bee. This year, she plans to win 11st place. She compiled a long list of words to study, 23%23\% of which have a Latin root.\newlineIf Ashley randomly chooses a word to study from the list 55 different times this month, what is the probability that exactly 33 of the words have a Latin root?\newlineWrite your answer as a decimal rounded to the nearest thousandth.\newline____

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Q. Last year, Ashley came in 22nd place in her school's Spelling Bee. This year, she plans to win 11st place. She compiled a long list of words to study, 23%23\% of which have a Latin root.\newlineIf Ashley randomly chooses a word to study from the list 55 different times this month, what is the probability that exactly 33 of the words have a Latin root?\newlineWrite your answer as a decimal rounded to the nearest thousandth.\newline____
  1. Calculate Probability: Now, we need to calculate the probability of picking exactly 33 words with a Latin root out of 55 tries. This is a binomial probability problem, where we use the formula P(X=k)=(nk)⋅(pk)⋅((1−p)(n−k))P(X=k) = \binom{n}{k} \cdot (p^k) \cdot ((1-p)^{(n-k)}), where nn is the number of trials, kk is the number of successes, pp is the probability of success, and (nk)\binom{n}{k} is the binomial coefficient.
  2. Calculate Binomial Coefficient: Calculate the binomial coefficient for 55 choose 33. This is 5!3!×(5−3)!\frac{5!}{3! \times (5-3)!}, which is 1010.
  3. Calculate Probability of Success: Now, calculate the probability of getting exactly 33 words with a Latin root. Using the binomial probability formula, we get P(X=3)=(53)×(0.233)×(0.772)P(X=3) = \binom{5}{3} \times (0.23^3) \times (0.77^2).
  4. Plug in Numbers: Plug in the numbers: P(X=3)=10×(0.233)×(0.772)P(X=3) = 10 \times (0.23^3) \times (0.77^2).
  5. Calculate Powers: Calculate the powers: (0.233)=0.012167(0.23^3) = 0.012167 and (0.772)=0.5929(0.77^2) = 0.5929.
  6. Multiply Everything: Multiply everything together: P(X=3)=10×0.012167×0.5929P(X=3) = 10 \times 0.012167 \times 0.5929.
  7. Perform Multiplication: Perform the multiplication: P(X=3)=10×0.012167×0.5929=0.0721P(X=3) = 10 \times 0.012167 \times 0.5929 = 0.0721.

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