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Compound Probability
Compound probability is the probability of two or more independent events occurring together. Compound probability can be calculated for two types of compound events, namely, mutually exclusive and mutually inclusive compound events. The formulas to calculate the compound probability for both types of events are different.
Compound probability is a concept that is widely used in the finance industry to assess risks and assign premiums to various policies. In this article, we will learn more about compound probability, its formulas, how to determine it as well as see various associated examples.
1.  What is Compound Probability? 
2.  Compound Probability Formulas 
3.  Compound Probability Example 
4.  How to Find Compound Probability? 
5.  FAQs on Compound Probability 
What is Compound Probability?
The compound probability of compound events (mutually inclusive or mutually exclusive) can be defined as the likelihood of occurrence of two or more independent events together. An independent event is one whose outcome is not affected by the outcome of other events. A mutually inclusive event is a situation where one event cannot occur with the other while a mutually exclusive event is when both events cannot take place at the same time. The compound probability will always lie between 0 and 1.
Compound Probability Formulas
There are two formulas to calculate the compound probability depending on the type of events that occur. In general, to find the compound probability, the probability of the first event is multiplied by the probability of the second event and so on. The compound probability formulas are given below:
Mutually Exclusive Events Compound Probability
P(A or B) = P(A) + P(B)
Using set theory this formula is given as,
P(A ∪ B) = P(A) + P(B)
Mutually Inclusive Events Compound Probability
P(A or B) = P(A) + P(B)  P(A and B)
P(A ∪ B) = P(A) + P(B)  P(A ⋂ B)
where A and B are two independent events, and P(A and B) = P(A) x P(B)
Compound Probability Example
Suppose a coin is tossed. The outcome of getting heads will be a simple event with a probability of 1 / 2. However, if the coin is tossed twice then the outcome of getting two heads will be a compound event. The compound probability of this event can be calculated as (1 / 2) x (1 / 2) = 1 / 4 or 0.25. This is an example of compound probability.
How to Find Compound Probability?
The steps to apply the compound formulas can be understood with the help of an example. Suppose the probability of Ryan failing an exam is 0.3 and the probability of Berta failing is 0.2. Then to find the compound probability of Ryan or Berta failing, the steps are as follows:
 Determine if the compound event is mutually exclusive or inclusive. This is an example of a mutually inclusive event.
 List the given probabilities. P(R) = 0.3 and P(B) = 0.2.
 Determine the correct compound probability formula. This is P(A or B) = P(A) + P(B)  P(A and B) for the given example
 Find P(A and B) which is given by P(A) x P(B). Thus, 0.2 x 0.3 = 0.6.
 Plug the values into the formula to get the result. Thus, the compound probability for the example is 0.44
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Important Notes on Compound Probability
 Compound probability is the likelihood of occurrence of two independent compound events together.
 Compound probability can be calculated for mutually exclusive and mutually inclusive compound events.
 P(A or B) = P(A) + P(B) and P(A or B) = P(A) + P(B)  P(A and B) are the compound probability formulas.
Examples on Compound Probability

Example 1: There are 40 girls and 30 boys in a class. 10 girls and 20 boys like tennis while the rest like swimming. If a student is selected at random then what is the probability that it will be a boy or a girl.
Solution: If a student is selected it can only be a girl or a boy. Thus, the probability that the selected student will be a girl or a boy is 1.
Answer: P(Boy or Girl) = 1

Example 2: If a dice is rolled then find the compound probability that either a 2 or 3 will be obtained.
Solution: P(2) = 1 / 6
P(3) = 1 / 6
As this is an example of a mutually exclusive event thus, the compound probability formula used is
P(A or B) = P(A) + P(B)
P(2 or 3) = (1 / 6) + (1 / 6)
= 2 / 6
= 1 / 3
Answer: P(2 or 3) = 1 / 3

Example 3: Find the compound probability of selecting 5 or a black card from a deck.
Solution: Total number of cards in a deck = 52
Number of cards in each suit = 13
P(Black card) = 26 / 52
Total number of 5s in a deck = 4
P(5) = 4 / 52
Number of 5s in black cards = 2
P(Black card and 5) = 2 / 52
This is a mutually inclusive event thus, the compound probability formula is
P(Black card or 5) = P(Black card) + P(5)  P(Black card and 5)
= 28 / 52
= 7 / 13
Answer: P(Black card or 5) = 7 / 13
FAQs on Compound Probability
What is the Meaning of Compound Probability?
Compound probability, in probability and statistics, is the probability that describes the chance that two or more independent events will occur together. It is determined by multiplying the probabilities of the occurring events.
What is the Formula for Compound Probability?
There are two formulas available for calculating the compound probability. These are given as follows:
 P(A or B) = P(A) + P(B)
 P(A or B) = P(A) + P(B)  P(A and B)
What is the Formula for Compound Probability in Set Theory?
The compound theory formulas expressed using set operations are given as follows:
 P(A ∪ B) = P(A) + P(B)
 P(A ∪ B) = P(A) + P(B)  P(A ⋂ B)
What are the Types of Events Used in Compound Probability?
There are two types of events used in compound probability. These are mutually exclusive and mutually inclusive compound events.
Can Compound Probability be Greater Than 1?
The compound probability value will always lie between 0 and 1. 0 indicates that the event will never occur and 1 denotes that the event will definitely take place.
What is the Difference between Simple and Compound Probability?
Simple probability is used to give the likelihood that one event will take place. On the other hand, compound probability gives the probability of occurrence of more than one separate event.
How to Calculate Compound Probability?
The steps to calculate the compound probability are as follows:
 Determine if the event is mutually inclusive or mutually exclusive.
 Apply the corresponding formula.
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