Is P(A-B) = P(A) -P(B)Here p is the probability function

• crotical
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crotical
Is
P(A-B) = P(A) -P(B)

Here p is the probability function

If B is a subset of A then, yes. However, if B is not a subset of A, in which case "A- BZ means "all members of A that are not also in B", this is not necessarily true. For example, suppose we choose a number from 1 to 100, each number being equally likely. Let A be the set of all multiples of 3, Be the set of all multiples of 6. There are 100/3= 33 multiples of 3 so P(A)= 0.33. There are 100/6= 16 multiples of 6 so P(B)= 16/100= 0.16. Of course, all multiples of 6 are multiples of 3 so A- B is just all multiples of 3 that are NOT multiples of 6 and there are 33- 16= 17 such numbers. P(A- B)= 17/100= 0.17= 0.33- 0.16= P(A)- P(B).

But suppose that B, instead of being "multiples of 6" is "even numbers (multiples of 2)". We still have P(A)= 0.33 but now P(B)= 0.5. Of course it is impossible that "P(A- B)= P(A)- P(B)" because P(A)- P(B)= 0.33- 0.5= -0.17 and a probability cannot be negative. In fact, "A- B" would mean removing from A all those numbers that are even and multiples of 2 which is the same as "multiples of 6". For this A and B, we would still have P(A-B)= 0.17 which is, as I said, NOT "P(A)- P(B)".

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')

I took S here as universal set

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')

You can shorten it a bit: $A = A\cap[B \cup B'] = (A \cap B) \cup (A \cap B'),$ and these last two sets are mutually exclusive. Thus $P(A) = P(A \cap B) + P(A \cap B'), \text{ hence } P(A \cap B') = P(A) - P(A) \cdot P(B) = P(A) \cdot P(B').$

RGV

1. What is the meaning of "P(A-B) = P(A) -P(B) here p is the probability function"?

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.

2. Can you give an example of how to use this equation in a real-life scenario?

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.

3. Is this equation always true for any events A and B?

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).

4. How can we use this equation to simplify probability calculations?

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.

5. Are there any other important equations or concepts related to probability that I should know?

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.

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