Video Transcript
A bag contains three pink marbles, four orange marbles, and five yellow marbles. Two marbles are selected without replacement. Using a tree diagram, find the probability that the second marble is yellow given
that the first marble is not yellow.
In this question, we’re asked to draw a tree diagram. And our first thoughts might be to draw one with the three different colored marbles
— pink, orange, and yellow — as we know the first and second marble can be any of
those three colors. However, before doing this, it is worth considering that the question is only
interested in whether the marble is yellow or not. We want to find the probability that the second marble is yellow given that the first
marble is not yellow. As a result, it is more sensible to simply consider the probabilities of whether the
first and second marbles are yellow or not.
There are five yellow marbles out of a total of 12. Therefore, the probability that the first marble is yellow is five twelfths. The probability that this first marble is not yellow is seven twelfths, as these
probabilities sum to one. We could also work this out by adding the number of pink marbles to the number of
orange marbles. This is equal to seven. We are then selecting a second marble and will once again consider whether it is
yellow or not yellow. As the marble is not being replaced, we are dealing with dependent events.
The second event does depend on what happens in the first. If the first marble is yellow, there are now four yellow marbles left in the bag and
a total of 11 marbles. The probability that the second marble is yellow given that the first marble is
yellow is therefore equal to four elevenths. If the first marble is yellow, the probability that the second marble is not yellow
is seven elevenths, as there are seven marbles left in the bag that are not yellow,
and there are a total of 11 marbles altogether.
Once again, it is worth checking that the probabilities on these two branches sum to
one. We can repeat this process for when the first marble is not yellow. Once again, there are 11 marbles left in the bag, five of which this time are yellow
and six are not yellow. Therefore, the respective probabilities are five elevenths and six elevenths.
The words “given that” mean that we are dealing with conditional probability. We want to find the probability that the second marble is yellow given that the first
is not yellow. This can be found from the tree diagram by firstly following the branch where the
first marble is not yellow. As the second marble needs to be yellow, the branch that corresponds to this has a
probability of five elevenths.
We can therefore conclude that the probability that the second marble is yellow given
that the first marble is not yellow is five elevenths. Whilst it was not required for this question, it is also worth recalling our formula
for conditional probability of dependent events. This states that the probability of 𝐴 given 𝐵 is equal to the probability of 𝐴
intersection 𝐵 divided by the probability of 𝐵.
