Video Transcript
A bag contains two black balls and eight white balls. Isabella selects two balls without replacement and draws the following tree diagram. Find the values of 𝑎 and 𝑏. Hence, calculate the probability that the second ball is white, given that the first ball is white.
We begin by noting that as Isabella is selecting two balls without replacement, the events are dependent. The second ball that Isabella selects is dependent on the first ball. We can answer the first part of our question by simply recalling that the sum of the probabilities for each set of branches must equal one. This means that 𝑎 plus eight-ninths is equal to one. Subtracting eight-ninths from both sides of this equation, we have 𝑎 is equal to one-ninth. Likewise, 𝑏 plus two-ninths is equal to one. The value of 𝑏 is therefore equal to seven-ninths. The answers to the first part of our question are one-ninth and seven-ninths, respectively.
The second part of our question asks us to calculate the probability that the second ball is white, given that the first ball is white. This is an example of conditional probability and can be written as shown. We begin by following the branch that represents the first ball being white. We then need to follow the branch where the second ball is white. It is the value on this branch of our tree diagram that we are interested in. Therefore, the probability that the second ball is white, given that the first ball is white, is our value of 𝑏, seven-ninths.
Whilst it is not required in this question, an alternative method to answer the second part would be to use our formula for conditional probability. The probability of 𝐴 given 𝐵 is equal to the probability of 𝐴 intersection 𝐵 divided by the probability of 𝐵 for dependent events.
