# Tree Diagrams. How to draw a tree diagram for independent events (2 balls from a bag)

A tree diagram can be used to work out the outcomes and probabilities when an experiment is being repeated more than once, such as, tossing a coin 3 times in a row or choosing 2 balls out of a bag. In the examples that follow the events are independent events (the second event is not conditional on the first event). These are the simplest type of tree diagram as the probabilities are much easier to calculate.

Example

There are two bags of balls, bag A and bag B. In bag A there are 3 red balls and 4 blue balls. In bag B there are 6 red balls and 3 blue balls. A ball is chosen from each bag. By drawing a tree diagram, work out the probability of getting two balls the same colour.

First of all choose a ball from bag A which contains 3 red balls and 4 blue balls. Since there are 7 balls altogether in bag A then the probability of a picking a red is 3/7 and the probability of picking a blue is 4/7. Put these in the first set of branches of the tree diagram.

Next pick a ball from bag B which contains 6 red balls and 3 blue balls. In bag B there are 9 balls in total, so the probability of picking a red ball is 6/9 and the probability of a blue ball is 3/9. It doesn’t matter what colour you have picked out previously from bag A (as the events are independent) so these probabilities can be repeated down the second set of branches of the tree diagram.

One the tree diagram is completed you can now work out the probability of picking 2 balls the same colour. The two outcomes that you need to calculate are 2 red balls or 2 blue balls.

The probability of picking 2 red balls can be found by multiplying 3/7 and 6/9:

3/7 × 6/9 = 18/63

The probability of picking 2 blue balls can be found by multiplying 4/7 and 3/9:

4/7 × 3/9 = 12/63

Finally, if you add these probabilities together you will get the probability of picking two balls the same colour from the bags:

18/63 + 12/63 = 30/63 (or 10/12)

Extra Tips for drawing tree diagrams:

Make sure you put the probability in the centre of each branch and the outcome at the end of each branch.

Don’t simplify the fractions on your tree diagram as this will make following calculations harder. Simplify the fraction once you get to your final answer.

You can check that the probabilities are correct as the probabilities on each pair (or triple) of branches will add up to 1. Also if you calculate all of the outcomes from the tree diagram then these probabilities will also sum to 1.

Always multiply the probabilities along the branches to calculate each outcome (don’t add them!)