Tree Diagram & Conditional Probability

What Is Independent Probability And Conditional Probability?
Regarding mathematical probability, there are two types, namely, independent probability and dependent probability. Independent probability indicates that the probability of an event is not affected or determined by any other events. This means the change of other events won't affect the probability of the independent events. A good example of independent probability is tossing a coin in which head and tail both have 50% independent probability. No matter what the past results are, the probability of the next toss will always be 50-50. In comparison, the dependent probability, also called conditional probability, will be dependent, or affected by the previous events. It is just like a series of things linked together. The result from the previous events will affect the subsequent events. For example, if you have a bag having 2 red balls and 2 green balls in it. Every time you grab a ball from the bag, the ratio of remaining red balls and green balls will change. So, in the next round, the probability to get a specific colored ball will change. If you get a red ball in the first round, there will be 1 red ball and 2 green balls in the bag. So, next time, you will have a 1/(1+2) probability to get another red ball. On the other hand, if you grab a green ball in the first round, you will have a probability of 2/(1+2) to get a red ball in the second round.

Using A Tree Diagram to Represent A Conditional Probability
In most cases, we can simply use a tree diagram to illustrate a conditional probability. Let's revisit the previous red and green ball example. In the first round, you will get a 1/2 chance to get a red or green ball. If you get a red ball in the first round, we can go one step further. You will see we will have a probability of 1/3 to get another red ball and 2/3 probability to get a green ball. So, if we want to know what is the final probability to get two red balls in the first two rounds, we can get the result by multiplying the two probabilities, which will be 1/2 x 1/3 = 1/6. So, from a tree diagram, we can show each conditional probability on the branches of the tree. To get the final outcome for each branch, we can just multiply all the probabilities along the branches. If we want to get the probabilities of multiple branches, we can just add them together.

Let's see another example. Bob finished school today and he wants to have pasta for dinner. His dinner will be prepared by either his mum or his dad. Bob's mum likes cooking so she will have a 70% probability to prepare dinner for Bob tonight. Since she doesn't like pasta, there is only a 20% probability to cook pasta if she cooks tonight. On the other hand, Bob's dad has a 30% probability to cook dinner for Bob tonight. Different from Bob's mom, Bob's dad is a pasta lover and there is a probability of 80% he will cook pasta if he will prepare the dinner. So, the question is, what is the probability that Bob will have pasta for dinner tonight? We can put those conditions into a tree diagram and quickly get the result. The first level of the branch will be 70% for mom and 30% for dad, if we go one step further, we can put the probabilities to the second level as well. So, we can get the following result. If Bob's mom cooks dinner and Bob will have a probability of 70% x 20% = 14% to have pasta tonight. If Bob's dad cooks dinner and Bob will have a probability of 30% x 80% = 24% to have pasta. So, in total we can calculate Bob will have a probability of 14% + 24% = 38% to have pasta for dinner. In summary, when you are struggling with the calculation of some probabilities, a tree diagram may be helpful to understand things better.