Questions: An opaque bag contains 16 yellow marbles and 16 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

The bag contains 16 yellow marbles and 16 teal marbles. The total number of marbles is: \[ 16 + 16 = 32 \]

The number of ways to choose 2 marbles out of 32 is given by the combination formula: \[ \binom{32}{2} = \frac{32!}{2!(32-2)!} = \frac{32 \times 31}{2 \times 1} = 496 \]

There are 16 teal marbles. The number of ways to choose 2 teal marbles out of 16 is: \[ \binom{16}{2} = \frac{16!}{2!(16-2)!} = \frac{16 \times 15}{2 \times 1} = 120 \]

The probability \( P \) that both marbles chosen are teal is the ratio of the number of favorable outcomes to the total number of outcomes: \[ P = \frac{\text{Number of ways to choose 2 teal marbles}}{\text{Total number of ways to choose 2 marbles}} = \frac{120}{496} \]

Simplify the fraction and round to four decimal places: \[ P = \frac{120}{496} \approx 0.2419 \]

Final Answer