Questions: An opaque bag contains 10 yellow marbles and 18 teal marbles. If two marbles are randomly chosen from the bag at the same time, find the probability that both marbles are teal. Round your answer to four decimal places.

Solution Steps

We need to find the probability of drawing 2 teal marbles from a bag containing 10 yellow marbles and 18 teal marbles. The total number of marbles in the bag is \( N = 10 + 18 = 28 \).

In this scenario:

• Total number of marbles, \( N = 28 \)
• Total number of teal marbles (successes), \( K = 18 \)
• Number of marbles drawn, \( n = 2 \)
• Number of teal marbles we want in the sample, \( k = 2 \)

The probability of drawing exactly \( k \) successes (teal marbles) in \( n \) draws can be calculated using the hypergeometric distribution formula:

\[ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} \]

Substituting the values:

\[ P(X = 2) = \frac{\binom{18}{2} \binom{10}{0}}{\binom{28}{2}} \]

Calculating each binomial coefficient:

• \( \binom{18}{2} = \frac{18 \times 17}{2 \times 1} = 153 \)
• \( \binom{10}{0} = 1 \)
• \( \binom{28}{2} = \frac{28 \times 27}{2 \times 1} = 378 \)

Thus, we have:

\[ P(X = 2) = \frac{153 \times 1}{378} = \frac{153}{378} \approx 0.4048 \]

Final Answer