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This Friday, Tucker has a French vocabulary test as well as a Spanish vocabulary test. To prepare, he made a study card for each word, 44%44\% of which are French. Every time he picks a card, he sticks it back in the deck and shuffles again.\newlineIf Tucker picks a study card from the deck 55 times during his first study session, what is the probability that exactly 33 cards have a French word?\newlineWrite your answer as a decimal rounded to the nearest thousandth.\newline____\newline

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Q. This Friday, Tucker has a French vocabulary test as well as a Spanish vocabulary test. To prepare, he made a study card for each word, 44%44\% of which are French. Every time he picks a card, he sticks it back in the deck and shuffles again.\newlineIf Tucker picks a study card from the deck 55 times during his first study session, what is the probability that exactly 33 cards have a French word?\newlineWrite your answer as a decimal rounded to the nearest thousandth.\newline____\newline
  1. Find French Card Probability: First, we need to find the probability of picking a French card. Since 44%44\% of the cards are French, the probability of picking a French card is 0.440.44.
  2. Calculate Non-French Card Probability: The probability of picking a non-French card is then 1−0.441 - 0.44, which is 0.560.56.
  3. Apply Binomial Probability Formula: We use the binomial probability formula, which is 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 on a single trial, and (nk)\binom{n}{k} is the binomial coefficient.
  4. Calculate ((53))(5 \choose 3): For exactly 33 French cards, we have n=5n=5, k=3k=3, and p=0.44p=0.44. So we need to calculate ((53)×(0.443)×(0.56(5−3)))(5 \choose 3) \times (0.44^3) \times (0.56^{(5-3)}).
  5. Calculate 0.4430.44^3: Calculating ((53))(5 \choose 3) gives us 5!3!⋅(5−3)!=10\frac{5!}{3! \cdot (5-3)!} = 10.
  6. Calculate 0.5620.56^2: Now we calculate 0.4430.44^3 which is 0.0851840.085184.
  7. Multiply Results: Next, we calculate 0.5620.56^2 which is 0.31360.3136.
  8. Multiply Results: Next, we calculate 0.5620.56^2 which is 0.31360.3136. Multiplying these together, we get 10×0.085184×0.3136=0.267310 \times 0.085184 \times 0.3136 = 0.2673.

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