During your GCSE maths exam, you will be required to calculate the likelihood of various conditional probabilities:
1 - Representing conditional probabilities on tree diagrams
As mentioned in the revision guide "Probability Diagrams"; tree diagrams are used to demonstrate all of the possible outcomes of an event. According to the rules of tree diagrams:
- The sum of the probabilities for any set of branches in a tree diagram will always be 1.
- In a tree diagram, you can calculate the probability of an outcome by multiplying along the branches and adding vertically.
When calculating conditional probability problems, it is important to remember that, after each event the numerators and denominators of each individual outcome must change due to the fact that one object has been removed and not replaced:
(a) A bag contains 2 blue beads and 3 red beads.
(i) Create a tree diagram which demonstrates the probabilities if one bead is chosen from the bag and then a
second bead is chosen from the bag without the first bead being replaced.
(ii) Use your tree diagram to calculate the probability of the first bead being blue and the second bead being red.
(a) (i) There are a total of 5 beads in the bag; of which 2 are blue and 3 are red.
Therefore, the probability of the first bead being blue is 2/5 and the probability of the first bead being red is 3/5.
Now that one bead has been taken out of the bag, there are only 4 left remaining.
If the first bead taken out was blue, then there would be 1 blue bead and 3 red beads remaining. Therefore, the probability of the second bead being blue is 1/4 and the probability of the second bead being red is 3/4.
Alternatively, if the first bead taken out was red, then there would be 2 blue beads and 2 red beads remaining. Therefore, the probability of the second bead being blue is 2/4 and the probability of the second bead being red is 2/4.
This information can be represented in the following tree diagram: