Probability P(A\B) = P(A) - P(B)

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In summary: And then what is P(A \cap B) ?In summary, we are trying to prove that P(A \setminus B) = P(A) - P(B). By using the definition of set difference, we can rewrite P(A \setminus B) as P(A \cap B^{C}). Then, using the properties of probability, we can rearrange this to get P(A) - P(A \cap B). Finally, by recognizing that B is a subset of A, we can substitute P(A \cap B) with P(B) and simplify to get P(A) - P(B), which is equal to P(A \setminus B).
  • #1
magicarpet512
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Homework Statement


Let [itex]A \subseteq B \subseteq S[/itex] where [itex]S[/itex] is a sample space.
Show that [itex]P(A \setminus B) = P(A) - P(B)[/itex]


Homework Equations



[itex]A \setminus B[/itex] denotes set difference; these are probability functions.

The Attempt at a Solution


I have,
[itex]P(A \setminus B) = P(A \cap B^{C})
= P(A) - P(A \cap B)
= P(A) - [P(B) - P(A^{c} \cap B)]
= P(A) - P(B) + P(A^{c} \cap B)[/itex]

It seems like I'm close, but I've spent a while trying to figure out how to get rid of the [itex]P(A^{c} \cap B)[/itex].

Any insight anyone?
Thanks!
 
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  • #2
I think your inclusion is backwards: you mean [itex] B \subseteq A \subseteq S [/itex], right?

Try writing A as the union of two disjoint sets--this will give you P(A) in terms of something that can be rearranged into what you're trying to prove.
 
  • #3
spamiam said:
I think your inclusion is backwards: you mean [itex] B \subseteq A \subseteq S [/itex], right?

Yes, thank you.

spamiam said:
Try writing A as the union of two disjoint sets--this will give you P(A) in terms of something that can be rearranged into what you're trying to prove.

I've tried that... unless I'm missing something?
A as the union of disjoint sets is [itex]A = (A \cap B) \cup (A \cap B^{c})[/itex].
So, [itex]P(A) = P((A \cap B) + (A \cap B^{c})[/itex].
When i plug this in and do some rearranging, i just get right back to where i ended up in the original post?
 
  • #4
magicarpet512 said:
I've tried that... unless I'm missing something?
A as the union of disjoint sets is [itex]A = (A \cap B) \cup (A \cap B^{c})[/itex].
So, [itex]P(A) = P((A \cap B) + (A \cap B^{c})[/itex].
When i plug this in and do some rearranging, i just get right back to where i ended up in the original post?

Ah I see, your calculations just went off in an unexpected direction after the third equality. Take a look at your third equality: since [itex] B \subseteq A[/itex], then what is [itex] A \cap B[/itex]?
 

FAQ: Probability P(A\B) = P(A) - P(B)

What is the meaning of "Probability P(A\B) = P(A) - P(B)"?

The formula "Probability P(A\B) = P(A) - P(B)" represents the probability of event A occurring, given that event B has already occurred. It is known as the conditional probability, where the probability of A is adjusted based on the information that B has occurred.

How is the formula for conditional probability derived?

The formula for conditional probability is derived from the basic definition of probability, where the probability of an event is the number of favorable outcomes divided by the total number of outcomes. In the case of conditional probability, the total number of outcomes is reduced to only those outcomes where event B has occurred.

What is the relationship between conditional probability and independent events?

If two events A and B are independent, then the occurrence of one event does not affect the probability of the other event. In this case, the formula "Probability P(A\B) = P(A) - P(B)" reduces to "Probability P(A\B) = P(A)", as the probability of B has no effect on the probability of A.

Can the formula for conditional probability be applied to non-independent events?

Yes, the formula for conditional probability can be applied to any events, whether they are independent or not. However, it is important to note that the formula only works for events that are not dependent on each other, meaning the occurrence of one event does not affect the probability of the other event.

Are there any limitations to the formula for conditional probability?

One limitation of the formula for conditional probability is that it assumes that the events are mutually exclusive, meaning they cannot occur at the same time. If this assumption is not true, then the formula may not accurately represent the probability of event A given that event B has occurred.

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