Probability question involving intersections

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You can calculate the probability of A, B, and C occurring in any order and still get the same result. The correct formula is P(A)*P(B|A)*P(C|A and B), which is equivalent to your formula. There is no logical error in your reasoning.
  • #1
TheKracken
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Homework Statement


and is in reference to the intersection of a set.
I know the formula that P(A and B ) = P(B) * P(A|B)

Now we are given something of the nature of P(A and B and C) I am trying to figure out a formula for this. From my searches on google I only find people giving the formula and I really would like to figure it out for my self.

Here is my attempt even though it does not appear to align with the correct formula.

P(A and B and C) = P((A and B) and (C)
= P(K and C ) ; such that P(A and B) = P(K)
= P(C) * P(K|C) ( now I need to do ( B|A)
= P(C) *P(A and B|C)*P(B|A)

What exactly am I doing wrong? I must have made some sort of logical error.

Correct answer should be
P(A)*P(C|A and B) *P(B|A)
 
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  • #2
If A happens first then B and finally C,
P(C/K) = P(C and K)/P(K)
=>P(C and K) = P(K)*P(C/K)
= P(A and B)*P(C/A and B)

P(A and B) = P(A)P(B/A) because A happens first and then B takes place.
Formula is correct. In your formula,B happens first.
 
  • #3
TheKracken said:

Homework Statement


and is in reference to the intersection of a set.
I know the formula that P(A and B ) = P(B) * P(A|B)

Now we are given something of the nature of P(A and B and C) I am trying to figure out a formula for this. From my searches on google I only find people giving the formula and I really would like to figure it out for my self.

Here is my attempt even though it does not appear to align with the correct formula.

P(A and B and C) = P((A and B) and (C)
= P(K and C ) ; such that P(A and B) = P(K)
= P(C) * P(K|C) ( now I need to do ( B|A)
= P(C) *P(A and B|C)*P(B|A)

What exactly am I doing wrong? I must have made some sort of logical error.

Correct answer should be
P(A)*P(C|A and B) *P(B|A)

Note: this last expression is equal to
[tex] P(A)\, P(C|A \cap B)\, P(B | A) = P(C | A \cap B)\,\underbrace{ P(B|A)\,P(A)}_{=P(A \cap B)} \\
\hspace{2cm}= P(C | A \cap B) \,P(A \cap B) = P (C \cap A \cap B) [/tex]
You can go backwards and "undo" ##P(A \cap B \cap C) = P(C \cap A \cap B)## to get the final result you want.
 
Last edited:
  • #4
AdityaDev said:
If A happens first then B and finally C,
P(C/K) = P(C and K)/P(K)
=>P(C and K) = P(K)*P(C/K)
= P(A and B)*P(C/A and B)

P(A and B) = P(A)P(B/A) because A happens first and then B takes place.
Formula is correct. In your formula,B happens first.
The order of events is immaterial.
 

1. What is the intersection of two events in probability?

The intersection of two events in probability refers to the event where both events occur simultaneously. It is represented by the symbol "∩" and can also be thought of as the overlap between two events.

2. How do you calculate the probability of an intersection?

To calculate the probability of an intersection, you multiply the individual probabilities of the two events. This is known as the "intersection rule" in probability. For example, if the probability of event A is 0.5 and the probability of event B is 0.6, then the probability of the intersection of A and B is 0.5 x 0.6 = 0.3.

3. What is the difference between independent and dependent events in terms of intersections?

Independent events are events where the outcome of one event does not affect the outcome of the other event. In this case, the probability of the intersection is simply the product of the individual probabilities. Dependent events, on the other hand, are events where the outcome of one event does affect the outcome of the other event. In this case, the probability of the intersection is calculated using conditional probability.

4. Can the probability of an intersection be greater than 1?

No, the probability of an intersection cannot be greater than 1. This is because the intersection of two events represents the likelihood of both events occurring, and the maximum probability of an event occurring is 1. If the probability of an intersection is greater than 1, it means that the individual probabilities of the events were greater than 1, which is not possible.

5. How can intersections be used to solve real-world problems?

Intersections can be used to solve real-world problems by helping us calculate the probability of multiple events occurring together. This can be useful in fields such as finance, insurance, and risk management, where the likelihood of multiple events happening at the same time needs to be considered. For example, an insurance company can use intersections to calculate the probability of a customer making a claim for two different types of coverage at the same time.

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