Question

a bag contains 10 red marbles, 9 white marbles, and 10 blue marbles. you draw 3 marbles out at random, without replacement. what is the probability that all the marbles are red?
the probability that all the marbles are red is

what is the probability that exactly two of the marbles are red?
the probability that exactly two of the marbles are red is

what is the probability that none of the marbles are red?
the probability of picking no red marbles is

70.3 problem solving using binomial
probability
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Explanation:

Step1: Calculate total marbles

Total marbles = 10 (red) + 9 (white) + 10 (blue) = 29.

Step2: Probability all red (3 marbles)

We use combinations. The number of ways to choose 3 red marbles from 10 is \( \binom{10}{3} \), and the number of ways to choose 3 marbles from 29 is \( \binom{29}{3} \).
The formula for combinations is \( \binom{n}{k} = \frac{n!}{k!(n - k)!} \).
\( \binom{10}{3} = \frac{10!}{3!7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \)
\( \binom{29}{3} = \frac{29!}{3!26!} = \frac{29 \times 28 \times 27}{3 \times 2 \times 1} = 3654 \)
Probability (all red) = \( \frac{120}{3654} = \frac{20}{609} \approx 0.0328 \)

Step3: Probability exactly two red

We need to choose 2 red from 10 and 1 non - red from 19 (9 white + 10 blue).
Number of ways for 2 red: \( \binom{10}{2} = \frac{10!}{2!8!} = \frac{10 \times 9}{2 \times 1}=45 \)
Number of ways for 1 non - red: \( \binom{19}{1}=\frac{19!}{1!18!}=19 \)
Total favorable ways: \( 45\times19 = 855 \)
Probability (exactly two red) = \( \frac{855}{3654}=\frac{285}{1218}=\frac{95}{406}\approx0.234 \)

Step4: Probability none red (all non - red)

Non - red marbles = 19. Number of ways to choose 3 non - red marbles: \( \binom{19}{3}=\frac{19!}{3!16!}=\frac{19\times18\times17}{3\times2\times1}=969 \)
Probability (none red) = \( \frac{969}{3654}=\frac{323}{1218}\approx0.265 \)

Answer:

• Probability all marbles are red: \( \boldsymbol{\frac{20}{609}\approx0.0328} \)
• Probability exactly two marbles are red: \( \boldsymbol{\frac{95}{406}\approx0.234} \)
• Probability none of the marbles are red: \( \boldsymbol{\frac{323}{1218}\approx0.265} \)