Definition and examples of the Total Probability
This article will be about The total probability, how to use it with the formula and various examples.
¿What is the total probability?
The total probability is a method used in statistic that allows us to to find the probability of an outcome of which we do not know the favorable cases or anything in general about this outcome, but it is a complement of other outcomes, so, in cases like this what we have is that the probability is distributed between 2 or more parts. This is why this method is called “total probability” because we are going to find the general probability of an outcome that we have in parts.
To understand better the situations where we have to apply the total probability, we are going to see an example (The resolution of this example will be in the examples section of this page, to go there just scroll down): There is a discussion in a work group about the color that is going to represent the group and there are 2 colors which are the favorites. The 60% of the members of the group wants the black to be the color, while the other 40% think that the best color for the group is the green color, if we know that 40% of the black color and 30% of the green color are men ¿What is the probability that when we randomly choose one member of team, this is a man?
As we can see in the previous problem, the total percentage of the members of the group that are men is not given, but we know which percentage of each preference is a man, we know that 40% out of a 60% and a 30% out of a 40% is a man. So, to find the total probability of when choosing a random member this is a man we are going to use the total probability (Resolution to this problem in the examples section).
A really useful tool that will help us to work with the total probability is the tree diagram, this is used mainly to make a scheme about the problem and with this we can find the data easier just to multiply and sum the correct variables.
Formula of the total probability
The formula of the total probability uses the principle of the multiplication rule, what we do is multiply the probability that two or more events happen at the same time, for example, in the previous problem, if we want to find the probability that one of the members chooses the black color and this is also a man, we would have to multiply the percentage of the people that choose the black color by the percentage of men that choose the black color, and we would do the same thing with the green color, we multiply the people that choose the green color by the percentage of the men that choose the green color, and then we sum this two results and we would finally have the probability that one of the members is a man.
In a tree diagram we multiply the data in the same branch and at the end we sum the whole results of each branch.
- Total probability formula
- p(a) = Σ p(bi) * p(a | bi)
In this formula we are going must use probabilities between 0 and 1, not in percentages, therefore if the problem gives us the data in percentages we have to divide that percentage by 100 to make use of this formula.
Examples of total probability
Example 1: We are going to solve the example of this article: in a work group there is a decision to make about the color that will represent the team, 60% chooses the black color and out of this 40% are men, while the other 40% choose the green color and out of this 30% are men. Find the probability that is someone selects a random member of the group this is a man.
m = man b = black g = green
- First we are going to solve branch by branch and then we are going to sum the probabilities so the formula does not become confusing.
- men that choose black
- p(b) * p(m | b)
- 0.6 * 0.4
- 0.24
- men that choose green
- p(g) * p(m | g)
- 0.4 * 0.3
- 0.12
- And now we sum both results
- 0.24 + 0.12
- 0.36 * 100%
- 36%
Example 2: In a concessionaire of cars there are 3 types of cars: sedan (40%), van(50%) and sport cars(10%), if we know that the 10% of every type of car is red, if we choose one random car, what is the probability that this car is red.
s = sedan v = van sc = sport car r = red car
- p(s) * p(r | s)
- 0.4 * 0.1
- 0.04
- p(v) * p(r | v)
- 0.5 * 0.1
- 0.05
- p(sc) * p(r | sc)
- 0.1 * 0.1
- 0.01
- and now we sum the results
- 0.04 + 0.05 + 0.01
- 0.1 * 100%
- 10%
Example 3: A big amount of people attend to a church eventually of which the 65% of the people are women and 35% are men and the 30% of the men and the 60% of the women are married, what are the chances that if we ask anybody no matter the gender if he/she is married the answer is yes.
m = man w = woman mr = maried
- p(w) * p(mr | w)
- 0.65 * 0.6
- 0.39
- p(m) * p(mr | m)
- 0.35 * 0.3
- 0.105
- And now we find the total percentage of married people
- 0.39 + 0.105
- 0.495 * 100%
- 49.5%
Example 4: A company of music made a poll that asked about their favorite music gender, the 70% of the participants of this poll were teenagers and 30% were adults, if the 20% of the teenagers and the 80% of the adults said that their favorite gender of music was rock ¿What is the probability that if we choose one poll randomly the answer to the question is “rock”?
t = teenagers a = adults r = rock
- p(t) * p(r | t)
- 0.7 * 0.2
- 0.14
- p(a) * p(r | a)
- 0.3 * 0.8
- 0.24
- and we find the total percentage
- 0.14 + 0.24
- 0.38 * 100%
- 38%