Questions: Suppose a jar contains 7 red marbles and 31 blue marbles. If 2 marbles are randomly chosen from the jar at the same time, find the probability that both marbles are red. Round your answer to four decimal places. Question Help: Video Message instructor Submit Question

Solution Steps

To find the probability that both marbles are red, we need to calculate the probability of drawing 2 red marbles from the total marbles. First, calculate the total number of ways to choose 2 marbles from the jar. Then, calculate the number of ways to choose 2 red marbles. Finally, divide the number of favorable outcomes by the total number of outcomes and round the result to four decimal places.

The total number of marbles in the jar is given by the sum of red and blue marbles: \[ \text{total\_marbles} = 7 + 31 = 38 \]

The total number of ways to choose 2 marbles from 38 is calculated using the combination formula: \[ \text{total\_ways} = \binom{38}{2} = \frac{38 \times 37}{2} = 703 \]

The number of ways to choose 2 red marbles from 7 is: \[ \text{red\_ways} = \binom{7}{2} = \frac{7 \times 6}{2} = 21 \]

The probability \( P \) that both marbles chosen are red is given by the ratio of favorable outcomes to total outcomes: \[ P(\text{both red}) = \frac{\text{red\_ways}}{\text{total\_ways}} = \frac{21}{703} \approx 0.029871977240398292 \] Rounding this to four significant digits gives: \[ P(\text{both red}) \approx 0.0299 \]

Final Answer