5 Q) Probability Trees – Part 3
Conditional probability asks you to find the probability of something happening given that something else happens. Conditional probability questions usually use words like given.
Example 1
There are 8 balls in a bag; 3 of the ball are red and the rest are blue. Two balls are picked out of the bag at random and they are not replaced. Answer the following:
a) Draw the probability tree for this
b) Find the probability that a red ball is taken out of the bag given that the first ball taken out of the bag is a red ball
c) Find the probability that a blue is picked first given that a red ball is picked second
The first step to answer this question is to draw the tree diagram. We are told in the question that there is 8 ball in the bag, 3 of the balls are red and the rest of the balls are blue. This means that there are 5 blue balls in the bag (8 – 3 = 5).
We are able to draw the first branches on the probability tree. We can either pick a red or a blue ball. The probability of a red ball being picked is 3/8 and the probability of a blue ball being picked is 5/8. The first part of the probability tree is given below.
The question tells us that the ball that is picked is not replaced. This means that when we pick a ball the number of overall balls in the bag decreases by 1 (there will now be 7 balls in total) and the number of the coloured balls that we picked also decreases by 1. The number of coloured balls that we did not pick remains the same.
Let’s deal with the top part of the probability tree first, which is where a red ball has been picked first. As a red ball has been picked, it means that the number of red balls in the bag decreases by 1, which means that there are now 2 red balls in the bag. Also, the total number of balls decreases by 1; there are now 7 balls in the bag. The number of blue balls remains the same. This means that the probability of picking a red ball is 2/7 and the probability of picking a blue ball is 5/7. We can place these probabilities onto the probability tree.
The next step is to go back to the start and imagine that a blue ball has been picked first. As a blue ball has been picked, the number of blue balls in the bag decreases by 1, which means that there will be 4 blue balls. Also, the total number of balls decreases by 1; there are now 7 balls. No red balls were picked, which means that the number of red balls stay at 3. This means that when a blue ball is picked first, the probability of picking a red ball second is 3/7 and the probability of picking a blue ball second is 4/7. We can add these probabilities to the probability tree.
The next two parts of this question are all to do with conditional probabilities. It is considerable easier to work out conditional probabilities when we have the probabilities of each of the outcomes from the probability tree. Therefore, I am going to find these probabilities before answering the next two questions. We obtain the probabilities of the four outcomes by multiplying the probabilities on the branches. The probabilities are shown on the diagram below.
Part b
Part b asks us to find the probability that a red ball is taken out of the bag given that the first ball taken out of the bag is a red ball. Below is the generic formula and we are going to be modifying this formula to obtain the answer to this question.
When we modify this, the equation becomes:
We can find the probability of picking a red first by adding the probabilities of all of the outcomes that have a red first in them. The top two outcomes have a red being picked first; the top outcome has a red being picking first and second (RR), and the second outcome has a red being picked first and a blue being picked second (RB). We can add these probabilities to see what the probability of picking a red first is and this gives us a probability of 21/56 (6/56 + 15/56). We sub these values into the formula and then we type it into a calculator.
Therefore, the answer to part b is 2/7.
Part c
The probability tree is shown below so you don’t have to keep looking up.
Let’s now find out the answer to part c. Part c asks us to Find the probability that a red ball is picked first given that a blue ball is picked second. Again, we will be modifying the standard formula. The standard formula is:
And we modify it to become:
There is only one way where a red ball can be picked first and a blue ball picked second (RB) and the probability of this occurring is 15/56.
There are two ways that a blue ball can be picked second; these two ways are a red ball first then blue ball second (RB), and the second way is blue ball first and blue ball second (BB). We obtain the probability of a blue ball being picked second by adding the probability of these two ways together, which gives us 35/56 (15/56 + 20/56). We can sub these values into the above formula and type it into a calculator.
The answer to this question is 3/7.