Final answer:
The probability that the second ball selected is a blue ball (P(B2)) is 22/45, considering different scenarios whether a red or blue ball was drawn first.
Step-by-step explanation:
To find P(B2), the probability that the second ball selected is a blue ball, we need to consider that the first ball can be either blue or red. Let's calculate the probabilities associated with both scenarios.
- If a red ball is drawn first (event R1), there will be 5 red and 4 blue balls left in the urn. The probability of drawing a blue ball second in this case is 4/9, because we do not put the red ball back in the urn.
- If a blue ball is drawn first (event B1), it is replaced and another blue ball is added, so there will be 6 red and 6 blue balls in the urn. The probability of drawing a blue ball second in this case is 6/12, which simplifies to 1/2.
To find P(B2), we use the Law of Total Probability, summing over all the possible outcomes of the first ball:
P(B2) = P(B2|R1)P(R1) + P(B2|B1)P(B1)
Where:
- P(B2|R1) = 4/9 (Probability of drawing blue given red was drawn first)
- P(B2|B1) = 1/2 (Probability of drawing blue given blue was drawn first)
- P(R1) = 6/10 (Initial probability of drawing red)
- P(B1) = 4/10 (Initial probability of drawing blue)
Substitute the above probabilities into the formula:
P(B2) = (4/9)(6/10) + (1/2)(4/10) = 24/90 + 20/90 = 44/90
Therefore, the probability that the second ball selected is a blue ball is 44/90, which can be simplified to 22/45 when reduced to lowest terms.