- #1
crotical
- 10
- 0
Is
P(A-B) = P(A) -P(B)
Here p is the probability function
Please help
P(A-B) = P(A) -P(B)
Here p is the probability function
Please help
crotical said:Is this proof valid
Given P(A) and P(B) are independent , prove P(A) and P(B') independent too.
P(A∩B) = P(A)P(B)
P(A∩B) = P(A)P(S-B')
=P(A)(1-P(B'))
=P(A)-P(A)P(B')
P(A∩B)=P(A)-P(A)P(B')
P(A∩(S-B'))=P(A)-P(A)P(B')
P(A-A∩(B'))=P(A)-P(A)P(B')
P(A)-P(A∩B')=P(A)-P(A)P(B')
P(A∩B')=P(A)P(B')
This equation represents the probability of event A occurring without event B occurring. In other words, it is the probability of the intersection of A and the complement of B. The "p" in the equation represents the probability function, which assigns a probability value to each possible outcome.
Sure, let's say you are rolling a fair die. Event A is rolling an even number (2, 4, or 6) and event B is rolling a number greater than 4 (5 or 6). The probability of rolling an even number is 1/2 (P(A) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2) and the probability of rolling a number greater than 4 is also 1/2 (P(B) = 1/6 + 1/6 = 2/6 = 1/2). The probability of rolling an even number without rolling a number greater than 4 (A-B) is 1/3 (P(A-B) = 1/6 + 1/6 = 2/6 = 1/3). Therefore, P(A-B) = P(A) - P(B) = 1/3 - 1/2 = 1/6.
No, this equation is only true for independent events. If events A and B are dependent, meaning the occurrence of one event affects the probability of the other event, then this equation will not hold. In that case, we would use the formula P(A-B) = P(A) - P(A and B).
This equation is useful for calculating the probability of event A occurring without event B occurring, especially when the probability of event B is already known. Instead of having to calculate the probability of A and B occurring together and then subtracting that from the probability of A, we can simply use this equation.
Yes, there are many other important equations and concepts in probability, such as the addition rule, multiplication rule, and Bayes' theorem. It is also important to understand the difference between theoretical probability (based on mathematical calculations) and experimental probability (based on actual observations). Additionally, understanding concepts such as sample space, complementary events, and mutually exclusive events can also be helpful in probability calculations.