What is the Probability of Events A and B Occurring Simultaneously?

In summary, to find the probability of A intersect B, we use the formula P(A or B but not both) = P(A) + P(B) - 2P(A intersect B). By substituting the given values, we get 1/3 = 1/2 + 1/4 - 2P(A intersect B). Solving for P(A intersect B), we get P(A intersect B) = 1/12. Therefore, the probability of events A and B both occurring is 1/12.
  • #1
crays
160
0
Events A and B are events such that

P(A) = 1/2
P(B) = 1/4
P(A or B but not both) = 1/3.

Find P(A intersect B)

so far what i have in mind is that , for A to happens, it need to be 1/3 x 1/2 which is 1/6
and for B to happens, it need to be 1/4 x 1/3 which is 1/12. Then I'm stuck here, not sure how should i find the intersect point. Please help.
 
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  • #2
I'm not sure what you mean by "for A to happens, it need to be 1/3 x 1/2". Do you mean the probability that A happens is 1/6? No, you have already said that the probability that A happens is 1/2 alone.

Think about this simplified scenario: A contains 6 objects, 2 of which are also in B, and B contains 5 objects. |A|= 6, |B|= 5, |A intersect B|= 2. How many objects are in A union B (in "A or B")? If we just add |A|+ |B|= 11, that's two large because we are counting objects in A intersect B twice, once in A and once in B. There are |A|+ |B|- |A intersect B|= 6+ 5- 2= 9 objects in A union B. That's were we get the rule P(A union B)= P(A)+ P(B)- P(A intersect B). Now, to get the number of objects in "A or B but not both" we just subtract A intersect B again: 9- 2= 7.
Yes, that is correct: there are 6- 2= 4 objects in "A but not B", 5- 2= 3 objects in "B but not A" and so 4+ 3= 7 objects in A or B but not Both. The number of objects in A or B but not Both is |A|+ |B|- 2|A intersect B|.

Converting that to probability requires only dividing through by the total number of objects so: P(A or B but not both)= P(A)+ P(B)- 2P(A intersect B).

P(A or B but not both)= 1/3= 1/2+ 1/4- 2P(A intersect B).

Solve for P(A intersect B).
 
  • #4
crays said:
Events A and B are events such that

P(A) = 1/2
P(B) = 1/4
P(A or B but not both) = 1/3.

Find P(A intersect B)
P(A or B but not both) = P(A or B) - P(A and B)
P(A or B) = P(A) + P(B) - P(A and B)
P(A or B) - P(A and B) = P(A) + P(B) - 2P(A and B)

I think you can continue from here.
 

Related to What is the Probability of Events A and B Occurring Simultaneously?

1. What is event probability?

Event probability is the likelihood or chance that a specific event will occur. It is usually expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty.

2. How is event probability calculated?

Event probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This is known as the classical probability formula. Other methods, such as empirical and subjective probability, can also be used to calculate event probability.

3. What factors affect event probability?

Event probability can be affected by various factors, such as the number of possible outcomes, the likelihood of each outcome, and the presence of any influencing variables. For example, tossing a coin has a 50% probability of landing on heads, but if the coin is biased, this probability may be altered.

4. What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. Experimental probability, on the other hand, is based on actual observations and can vary depending on the sample size and the number of trials conducted.

5. How can event probability be used in real-life situations?

Event probability is used in various fields, such as finance, insurance, and sports. It can help make informed decisions by predicting the likelihood of certain events, such as stock market fluctuations, insurance claims, and sports outcomes. It is also used in risk assessment and decision-making processes.

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