Thread: Independent events

In independent events, what does "while presuming that B occurs, the probability that A occurs is" |A and B| / |B| mean.

Does this mean that first one dice is thrown (considering the case of two dices) and then what is the probability that the second dice will have one of the same outcome than that of the first one.

CB

Originally Posted by computer-bot

In independent events, what does "while presuming that B occurs, the probability that A occurs is" |A and B| / |B| mean.

Does this mean that first one dice is thrown (considering the case of two dices) and then what is the probability that the second dice will have one of the same outcome than that of the first one.

CB

It means that given that B has occured then the probability that A occurs is
P(A and B)/P(B). This may be written:

P(A|B)P(B) = P(A and B)

Also you do not need to assume that A and B are independent.

Suppose we are talking about the result of throwing a single die, and let
A be the event that an even number is thrown, and B be the event that
a number greater than 2 is thrown. Then P(A)=1/2, P(B)=4/6, P(A and B)=2/6
and P(A|B) is clearly 2/4 (there are 4 possible faces which show given that
B has occured of which 2 are even), and the formula tells us that:

P(A|B)=P(A and B)/P(B)=(2/6)(6/4)=2/4.

This is essentialy Bayes theorem, since:

P(A and B)=P(A|B)P(B)=P(B|A)P(A),

so:

P(A|B)=P(B|A)P(A)/P(B).

RonL

What does this expression mean?
P(A|B)

CB

That means probability of A given B.

Suppose you draw a card from a deck. What's the probability it's a King?.

Well, you would have 4/52. But, if you didn't replace the card and drew another, what would be the probability of another King. 3/51.

See, the probability of the second King was changed due to the first card being drawn. A given B.

If you had replaced the first card and drew from a deck that had 52 cards instead of 51, it would'nt have been dependent on the other card. See?.

Does that make sense?.

I would just like to make the observation that the example of independent events...

Event A. The dice shows an even number.
Event B. The same dice on the same throw shows a number greater than two.

is a good example since it shows that the independent events can be about the same "experiment". Books frequently give "the dice is a three" and "the coin is a tail" as their example of independent events, which is insufficient as it relies on physical separateness and not on the mathematical angle.

In my opinion the best way to think about independent events is to say to oneself, "If I know that P has occurred, does it affect my belief in the likelihood of Q?". If the answer is "no" then the events are independent.

Originally Posted by ray_sitf

In my opinion the best way to think about independent events is to say to oneself, "If I know that P has occurred, does it affect my belief in the likelihood of Q?". If the answer is "no" then the events are independent.

May I ask why anyone would introduce belief into any axiomatic system?

Originally Posted by Plato

May I ask why anyone would introduce belief into any axiomatic system?

Ahh.. Very good question.

(proceeding to over generalise:...)

We have an axiomatic system called probability theory, but we also have
a theory of rational behaviour under uncertainty (which may or may not be
how humans behave but that is beside the point) which has an element
modelling how uncertainty is handled which obeys the axioms of the formal
theory. This commonly called Bayesian probability and deals with degrees
of belief.

As far as I can see probability theory as a branch of measure theory would
be of no interest if it was not for this link to subjective probability, and the
incredible power of Bayesian statistics. (OK it is also of interest because of
the very mysterious appearance of probability in QM, and somewhat less
mysterious appearance in statistical mechanics)

RonL

(Dons flame retardant underwear and retires)

Originally Posted by ray_sitf

I would just like to make the observation that the example of independent events...

Event A. The dice shows an even number.
Event B. The same dice on the same throw shows a number greater than two.

is a good example since it shows that the independent events can be about the same "experiment". Books frequently give "the dice is a three" and "the coin is a tail" as their example of independent events, which is insufficient as it relies on physical separateness and not on the mathematical angle.

In my opinion the best way to think about independent events is to say to oneself, "If I know that P has occurred, does it affect my belief in the likelihood of Q?". If the answer is "no" then the events are independent.

Originally Posted by galactus

That means probability of A given B.

Suppose you draw a card from a deck. What's the probability it's a King?.

Well, you would have 4/52. But, if you didn't replace the card and drew another, what would be the probability of another King. 3/51.

See, the probability of the second King was changed due to the first card being drawn. A given B.

If you had replaced the first card and drew from a deck that had 52 cards instead of 51, it would'nt have been dependent on the other card. See?.

Does that make sense?.

Thanks for such a through discussion and ray_sitf, your example was brilliant. But let ask one more thing that the expression "Probability of A given B" is use only for independent events or just any event; or should I say that it could be used to check whether two events are independent or not and is just used to find the probability of two successive events.

CB

Originally Posted by computer-bot

Thanks for such a through discussion and ray_sitf, your example was brilliant. But let ask one more thing that the expression "Probability of A given B" is use only for independent events or just any event; or should I say that it could be used to check whether two events are independent or not and is just used to find the probability of two successive events.

CB

The definition of independence for events A and B is that:

P(A and B) = P(A) P(B).

In the example:

Event A. The dice shows an even number.
Event B. The same dice on the same throw shows a number greater than two.

P(A)=1/2, and P(B)=4/6=2/3, P(A and B)=1/3, so these are indeed independent events.

But now suppose that instead

Event B: The same dice on the same throw shows a number greater than three.

Now P(A)=1/2, P(B)=1/2, and P(A and B)=1/3, so now A and B are not independent, but still:

P(A and B) = P(A|B)P(B) = (2/3)(1/2) = 1/3.

As I tried to point out before "P(A and B) = P(A|B)P(B)" does not require
independence for A and B.

RonL