A bag contains three blue balls and seven red balls. David selects two balls without replacement and draws the following tree diagram. Given that the first ball is red, find the value of 𝑥 that represents the probability that the second ball selected is red.
As we are told that David selects the two balls without replacement, we are dealing with conditional probability. This means that the selection of the first ball will impact the selection of the second ball. We are asked to find the value of 𝑥, which is the probability that the second ball selected is red given that the first ball is red. This can be written using the conditional probability notation, as shown. As there were 10 balls initially in the bag, once David has selected one, there are nine balls remaining. There were seven red balls initially. And if David’s first selection is a red ball, there will be six red balls remaining in the bag. The probability that the second ball selected is red, given that the first ball is red, is six out of nine or six-ninths.
As both the numerator and denominator of the fraction are divisible by three, this simplifies to two-thirds. The value of 𝑥 on the tree diagram is two-thirds. At this stage, it is worth recalling that the sum of probabilities for each set of branches should equal one. Three-tenths plus seven-tenths equals one. Two-ninths plus seven-ninths equals one. And one-third plus our value of 𝑥, two-thirds, equals one. This is a useful check to complete after any question on tree diagrams.