Questions: A box contains 5 purple marbles, 3 green marbles and 2 orange marbles. Draws are made without replacement. P(both marbles are purple) 1 / 4 2/9 1 / 3 1 / 2

Solution Steps

To find the probability that both marbles drawn are purple, we need to consider the probability of drawing a purple marble on the first draw and then another purple marble on the second draw, without replacement. First, calculate the probability of drawing a purple marble on the first draw. Then, calculate the probability of drawing another purple marble given that the first one was purple. Multiply these probabilities to get the final answer.

The total number of marbles in the box is the sum of purple, green, and orange marbles: \[ 5 + 3 + 2 = 10 \]

The probability of drawing a purple marble on the first draw is the number of purple marbles divided by the total number of marbles: \[ P(\text{first purple}) = \frac{5}{10} = \frac{1}{2} \]

After drawing one purple marble, there are 4 purple marbles left and a total of 9 marbles remaining. The probability of drawing a second purple marble is: \[ P(\text{second purple} \mid \text{first purple}) = \frac{4}{9} \]

The probability of both marbles being purple is the product of the probabilities of each event: \[ P(\text{both purple}) = P(\text{first purple}) \times P(\text{second purple} \mid \text{first purple}) = \frac{1}{2} \times \frac{4}{9} = \frac{2}{9} \]

Final Answer