select the correct answer from each drop-down menu.

betty picks 4 random marbles from a bowl containing 3 white, 4 yellow, and 5 blue marbles.

the probability that exactly 1 of the 4 marbles drawn is blue is choose an answer. the probability that at least 1 of the 4 marbles drawn is white is choose an answer.

Question

select the correct answer from each drop-down menu.

betty picks 4 random marbles from a bowl containing 3 white, 4 yellow, and 5 blue marbles.

the probability that exactly 1 of the 4 marbles drawn is blue is choose an answer. the probability that at least 1 of the 4 marbles drawn is white is choose an answer.

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Combinations", "Classical Probability", "Hypergeometric Probability" ], "new_concepts": [], "current_concepts": [ "Combinations", "Classical Probability", "Hypergeometric Probability" ] } ## Calculate the total number of ways to choose 4 marbles $$ \begin{aligned} \text{Total marbles} &= 3 \text{ (white)} + 4 \text{ (yellow)} + 5 \text{ (blue)} = 12 \ \text{Total outcomes} &= \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \end{aligned} $$ ## Calculate the probability of drawing exactly 1 blue marble $$ \begin{aligned} \text{Favorable outcomes} &= \binom{5}{1} \times \binom{7}{3} = 5 \times \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 5 \times 35 = 175 \ P(\text{exactly 1 blue}) &= \frac{175}{495} \approx 0.3535 \end{aligned} $$ ## Calculate the probability of drawing at least 1 white marble $$ \begin{aligned} \text{Complement outcomes (0 white)} &= \binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \ P(\text{0 white}) &= \frac{126}{495} \approx 0.2545 \ P(\text{at least 1 white}) &= 1 - P(\text{0 white}) = 1 - \frac{126}{495} = \frac{369}{495} \approx 0.7455 \end{aligned} $$ ## Question 1 The probability that exactly 1 of the 4 marbles drawn is blue is 0.3535. ## Question 2 The probability that at least 1 of the 4 marbles drawn is white is 0.7455. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Hypergeometric Probability" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the total number of ways to choose 4 marbles

Calculate the probability of drawing exactly 1 blue marble

Calculate the probability of drawing at least 1 white marble

Answer:

Question 1

Question 2