What is the relationship between accidents and probability?

[balleand]

Ok so i was thinking this:

Since we all know that there is a certain probability that two accidents can happen simultaneously. That must mean that if I expose myself to a lot of accidents, I will increase the chances of your house burning down.

Is this logical?

 

[Tac-Tics]

Ok so i was thinking this:

Since we all know that there is a certain probability that two accidents can happen simultaneously. That must mean that if I expose myself to a lot of accidents, I will increase the chances of your house burning down.

Is this logical?

If two events are independent, then no, it doesn't increase the chances of your house burning down.

However, if I am prone to setting my shoes on fire and I'm a good friend of yours, it might increase the chances, because the events are now dependent.

But talking about accidents make it difficult, because there is so much ambiguity. It's easier to talk about coins or dice until you grasp the basic concepts.

An example of flipping coins is this:

If I flip a coin twenty times, we expect that we will get roughly 10 heads and 10 tails.

However, let's say I flip the first ten coins and, miraculously, they all end up heads. It is very unlikely that the next ten will all be tails. Given the first ten flips all ending in heads, we should expect roughly 15 heads and 5 tails at the end.

The reason is that all the events are independent of each other. Flipping one coin has no bearing on flipping another. They don't affect each other. Flipping ten coins and having some unlikely thing happen doesn't make a second very unlikely thing likely all of a sudden. In the case of the unlikely, Nature forgets its debts.

On the other hand, think about how our expected outcomes evolve over time. At the start, we expect 10-10. But half way through, we expect 15-5. Why are these expectations different? The answer is that the second expectation depends on the first five flips. We call this "conditional probability". Unlike the individual coin flips, the total heads and tails depend on ALL the flips. If we have some additional information about the flips, it changes the probability.

Going back to fuzzy insurance land, if I am known to visit my friends house and I am known to like juggling fire, then the two accidents could be dependent on each other, and the conditional probability (under the "condition" that I visit you and juggle fire) increases.

But again, dealing with stories about accidents, it's not always clear when two events are independent and when they aren't.

 

[balleand]

If two events are independent, then no, it doesn't increase the chances of your house burning down.

However, if I am prone to setting my shoes on fire and I'm a good friend of yours, it might increase the chances, because the events are now dependent.

But talking about accidents make it difficult, because there is so much ambiguity. It's easier to talk about coins or dice until you grasp the basic concepts.

An example of flipping coins is this:

If I flip a coin twenty times, we expect that we will get roughly 10 heads and 10 tails.

However, let's say I flip the first ten coins and, miraculously, they all end up heads. It is very unlikely that the next ten will all be tails. Given the first ten flips all ending in heads, we should expect roughly 15 heads and 5 tails at the end.

The reason is that all the events are independent of each other. Flipping one coin has no bearing on flipping another. They don't affect each other. Flipping ten coins and having some unlikely thing happen doesn't make a second very unlikely thing likely all of a sudden. In the case of the unlikely, Nature forgets its debts.

On the other hand, think about how our expected outcomes evolve over time. At the start, we expect 10-10. But half way through, we expect 15-5. Why are these expectations different? The answer is that the second expectation depends on the first five flips. We call this "conditional probability". Unlike the individual coin flips, the total heads and tails depend on ALL the flips. If we have some additional information about the flips, it changes the probability.

Going back to fuzzy insurance land, if I am known to visit my friends house and I am known to like juggling fire, then the two accidents could be dependent on each other, and the conditional probability (under the "condition" that I visit you and juggle fire) increases.

But again, dealing with stories about accidents, it's not always clear when two events are independent and when they aren't.

I get what your saying, but accidents was just an example. Let me try to explain my logic more in depth.

We know there is a chance of president Obama having a heart attack while I'm sitting in a chair right?
Lets call the probability of Obama having a heart attack "X"
and the probability of him having a heart attack while I'm sitting in a chair "Y"
That must mean that because we know that there is a chance of him having a heart attack while I'm sitting in a chair and also there is a chance of him having a heart attack while I'm not sitting in a chair the chance of him having a heart attack is X+Y
same goes for Obama having a heart attack while I'm walking (lets call it "Z")
the chance of him having a heart attack now is X+Y+Z
etc.

This could go on for ever, so ultimately the chance of him having a heart attack seems pretty high, no?

What i really want to know is weather or not my logic is flawed here. and if it is what are the missing pieces to my puzzle?

 

[statdad]

"We know there is a chance of president Obama having a heart attack while I'm sitting in a chair right?
Lets call the probability of Obama having a heart attack "X"
and the probability of him having a heart attack while I'm sitting in a chair "Y"
That must mean that because we know that there is a chance of him having a heart attack while I'm sitting in a chair and also there is a chance of him having a heart attack while I'm not sitting in a chair the chance of him having a heart attack is X+Y"

No: as events, Y and X are not disjoint, rather Y is a subset of X.

"same goes for Obama having a heart attack while I'm walking (lets call it "Z")
the chance of him having a heart attack now is X+Y+Z
etc."
Same problem here.

"This could go on for ever, so ultimately the chance of him having a heart attack seems pretty high, no?

What i really want to know is weather or not my logic is flawed here. "
Yes, it is - see points made above.

"and if it is what are the missing pieces to my puzzle?"
Poor framing of the events to start.

 

[Tac-Tics]

Lets call the probability of Obama having a heart attack "X"
and the probability of him having a heart attack while I'm sitting in a chair "Y"
That must mean that because we know that there is a chance of him having a heart attack while I'm sitting in a chair and also there is a chance of him having a heart attack while I'm not sitting in a chair the chance of him having a heart attack is X+Y...

This doesn't follow.

Let's leave poor Obama out of this example (Politics isn't my forte).

Let's talk about dice. I roll one. You roll one.

Let event X be when you roll a 6. The probability of X is 1/6.
Let event Y be when you roll a 6 and I roll an even number. The probability of Y is 1/36.

The chance of X and Y happening together is 1/36. It is NOT the probability of X plus the probability of Y (which would be 7/36). Because event X is not independent of event Y. In fact, when Y happens, X is guaranteed to happen. Y is a "subevent" of X.

You can only add probabilities if two events are independent. In the political version you gave, they are not.

 

[statdad]

"You can only add probabilities if two events are independent."

Disjoint, not independent.

 

[balleand]

"You can only add probabilities if two events are independent."

Disjoint, not independent.

alright so the probability of obama having a heart attack is X*Y*Z etc..?
now I am no longer adding but mutiplying, is that allowed?

 

[statdad]

If two events are disjoint then

$$P(A \cup B) = P(A) + P(B)$$

If two events are independent then

$$P(A \cap B) = P(A) \cdot P(B)$$

Your events are neither disjoint nor independent - you can't do the calculations you seem to want in the way you're trying to.

 

[Office_Shredder]

A simple example. You flip a coin. The probability of you getting a heads is 1/2. The probability of you not getting a tails is 1/2. But the probability of you either getting a heads or not getting a tails is not going to be 1/2+1/2=1, it's just going to be 1/2 still. The reason why is because these two events: getting a heads, and not getting a tails, are not disjoint, i.e. if one of them happens, the other one can happen (in this case, always happens).

The probability of getting a heads AND getting a tails is not 1/2*1/2=1/4, it's just 1/2. The reason why is because the events of getting a heads and not getting a tails are not independent, i.e. knowing that one happened changes the probability of the other one happening. Compare this to flipping two coins: knowing that one of them landed on heads does not tell you anything about the other coin

 

[balleand]

A simple example. You flip a coin. The probability of you getting a heads is 1/2. The probability of you not getting a tails is 1/2. But the probability of you either getting a heads or not getting a tails is not going to be 1/2+1/2=1, it's just going to be 1/2 still. The reason why is because these two events: getting a heads, and not getting a tails, are not disjoint, i.e. if one of them happens, the other one can happen (in this case, always happens).

The probability of getting a heads AND getting a tails is not 1/2*1/2=1/4, it's just 1/2. The reason why is because the events of getting a heads and not getting a tails are not independent, i.e. knowing that one happened changes the probability of the other one happening. Compare this to flipping two coins: knowing that one of them landed on heads does not tell you anything about the other coin

yeah...I get it now, must be frustrating to explain something to someone over and over, but yeah flipping coins is an easier concept to grasp than heart attacks and burning houses.

thanks for setting me straight though xD