Principle of inclusion-exclusion

[nicholaskong100]

I notice circling OR in addition to circling AND helps with fitting the givens into the principle of inclusion-exclusion formula.

I have several questions:

1) Can "AND" be represented as "operations"?
2) Can "OR" be represented as an "operations"?
3) Is A U B which represents two events, the sample space, does A U B = 1?

 

[Dr.Peterson]

View attachment 28591

I notice circling OR in addition to circling AND helps with fitting the givens into the principle of inclusion-exclusion formula.

I have several questions:

1) Can "AND" be represented as "operations"?
2) Can "OR" be represented as an "operations"?
3) Is A U B which represents two events, the sample space, does A U B = 1?

In this problem, OR means union; AND is just a bit of grammar!

You're told two things:

1. The probability of winning OR tying is 40%.
2. The probability of winning is 30%.

The word "and" just connects those two statements, saying they are both true.

A U B means win OR tie. So P(A ⋃ B) is not 1; it's 0.40. You are not told that P(A) = 0.40.

You also are not told that P(A ⋂ B) = 0.30. Think about it: Can you both win AND tie the same game?

Also, be very careful with words and symbols. A U B is an event, namely A or B. You can't say that A U B = 1; rather, P(A U B) = 1. The probability P(A U B) is a number.

 

[nicholaskong100]

In this problem, OR means union; AND is just a bit of grammar!

You're told two things:

1. The probability of winning OR tying is 40%.
2. The probability of winning is 30%.

The word "and" just connects those two statements, saying they are both true.

A U B means win OR tie. So P(A ⋃ B) is not 1; it's 0.40. You are not told that P(A) = 0.40.

You also are not told that P(A ⋂ B) = 0.30. Think about it: Can you both win AND tie the same game?

Also, be very careful with words and symbols. A U B is an event, namely A or B. You can't say that A U B = 1; rather, P(A U B) = 1. The probability P(A U B) is a number.

Do I ignore A intersects B since winning and tying a game cannot happen at the same time? How come it would be part of the Principle of inclusion exclusion formula?

I cannot seems to solve for B when I see .10 = | B | - | .30 and B| .

Am I making the math more complicated than it needs to be?

 

[Dr.Peterson]

Do I ignore A intersects B since winning and tying a game cannot happen at the same time? How come it would be part of the Principle of inclusion exclusion formula?

You aren't ignoring it; you are using the fact that P(A ⋂ B) = 0 in this case. As far as I can see, that is how you are to use the formula.

I cannot seems to solve for B when I see .10 = | B | - | .30 and B| .

| .30 and B| is meaningless. You are again confusing probabilities and events. They are not the same thing.

I'm also confused by your notation; this form of the equation is about cardinalities of sets, not probabilities. Is that what you were told to use?

But when you use P(A ⋂ B) = 0, everything will become simple.

 

[nicholaskong100]

You aren't ignoring it; you are using the fact that P(A ⋂ B) = 0 in this case. As far as I can see, that is how you are to use the formula.


| .30 and B| is meaningless. You are again confusing probabilities and events. They are not the same thing.

I'm also confused by your notation; this form of the equation is about cardinalities of sets, not probabilities. Is that what you were told to use?

But when you use P(A ⋂ B) = 0, everything will become simp

I decided to rewrite my solution given your feedback.

I actually was not given any tips to use the PIE formula. But the section prior to the questions talked about PIE. It made sense that PIE formula would be the only formula to use. And the other reason would be this was the only question that used the OR keyword which tipped me off.

When you say it is meaningless, do you mean it is not something we need to know to solve the problem because P(A intersects B) serves no purpose?

Would it have been possible to solve the problem through observation (looking at winning game is 30% compared to winning a game or tying a game is 40%) instead of using the PIE formula?

 

[Dr.Peterson]

When you say it is meaningless, do you mean it is not something we need to know to solve the problem because P(A intersects B) serves no purpose?

No, I meant exactly what I said: it means nothing. You can't intersect a number and an event; they are not the same kind of thing. P(A intersect B) or P(A and B) or P(A ⋂ B) means something; P(0.30 ⋂ B) is nonsense! Presumably you meant something else, but what you wrote means nothing.

Would it have been possible to solve the problem through observation (looking at winning game is 30% compared to winning a game or tying a game is 40%) instead of using the PIE formula?

Yes. I would make a Venn diagram; that is one way to make this observation. The 30% is part of the 40%.

The general formula is P(A U B) = P(A) + P(B) - P(A ⋂ B); in the special case of mutually exclusive events, where P(A ⋂ B) = 0, it becomes P(A U B) = P(A) + P(B). Have you been taught anything like this yet?

 

[nicholaskong100]

No, I meant exactly what I said: it means nothing. You can't intersect a number and an event; they are not the same kind of thing. P(A intersect B) or P(A and B) or P(A ⋂ B) means something; P(0.30 ⋂ B) is nonsense! Presumably you meant something else, but what you wrote means nothing.


Yes. I would make a Venn diagram; that is one way to make this observation. The 30% is part of the 40%.

The general formula is P(A U B) = P(A) + P(B) - P(A ⋂ B); in the special case of mutually exclusive events, where P(A ⋂ B) = 0, it becomes P(A U B) = P(A) + P(B). Have you been taught anything like this yet?

I'm teaching myself for the time being. I don't start the first day of the actual Probability and Statistics class until next week. I cannot seem to find a section in my text book where it is shown where P(A intersects B ) = 0.

Principle of inclusion and exclusion is only talk about in 2 pages: Page 11, Page 13.



Venn diagram is mentioned but only for one page. But it is written in terms of complements. Just curious, would A U B = (A intersects B's complement) U (A's complement intersects B)?

 

[Dr.Peterson]

Venn diagram is mentioned but only for one page. But it is written in terms of complements. Just curious, would A U B = (A intersects B's complement) U (A's complement intersects B)?

No, the Venn diagram shows that A U B = (A ⋂ B^c) U (A^c ⋂ B) U (A ⋂ B)

I cannot seem to find a section in my text book where it is shown where P(A intersects B ) = 0.

Does it use the terms "mutually exclusive" or "disjoint" anywhere? That's the name for this situation.