The probability of Raina picking alternating colors can be calculated by considering the two possibilities for the first ball she draws: orange or purple.
Scenario 1: Orange first (Orange - Purple - Orange)
Probability of picking an orange ball first: 3 out of 8 total balls (since there are 3 orange balls and 8 total at the beginning).
Probability of picking a purple ball second: 5 out of 7 remaining balls (after taking out the orange ball, there are 7 left, and 5 of them are purple).
Probability of picking an orange ball third: 2 out of 6 remaining balls (after taking out the purple ball, there are 6 left, and 2 of them are orange).
So, the probability for this scenario is (3/8) * (5/7) * (2/6) = 5/56.
Scenario 2: Purple first (Purple - Orange - Purple)
Similarly, we can find the probability for picking purple first, then orange, and then purple again.
Probability of picking a purple ball first: 5 out of 8 total balls.
Probability of picking an orange ball second: 3 out of 7 remaining balls.
Probability of picking a purple ball third: 4 out of 6 remaining balls.
The probability for this scenario is (5/8) * (3/7) * (4/6) = 5/56.
Total Probability
Since these two scenarios (orange first and purple first) are mutually exclusive (they can't happen at the same time), to get the total probability of picking alternating colors, we simply add the probabilities of each scenario:
Total Probability = (5/56) + (5/56) = 10/56 = 5/28
Therefore, the probability of Raina picking alternating colored balls is 5/28.
