Чи не повинна спільна ймовірність двох незалежних подій дорівнювати нулю?

Якщо спільна ймовірність є перетином двох подій, то чи не повинна спільна ймовірність двох незалежних подій дорівнювати нулю, оскільки вони взагалі не перетинаються? Я збентежений.

probability joint-distribution

— гастон
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Ймовірність того, що я дивлюсь телевізор в день, становить 1/2. Імовірність того, що в цей день дощить, дорівнює 1/2. Це незалежні події. Яка ймовірність того, що я дивлюся телевізор у дощовий день?

— користувач1936752

@ user1936752 Власне кажучи, ваші приклади подій для більшості людей не є незалежними (наприклад, вони можуть бути більш охочими проводити час на свіжому повітрі, коли не йде дощ)

— Хаген фон Ейтцен

@HagenvonEitzen OK, good point. Change rainy day to eat chocolate.

— Rui Barradas

@Gaston: Don't confuse "independent" with "mutually exclusive". Independent events are completely unrelated to each other, whereas mutually exclusive events are inherently related. For example, suppose I flip two coins: whether I get heads on Coin 1 is unaffected by the result of Coin 2, but it's inherently connected to whether I get tails on Coin 1! =)

— jdmc

This video here and this other one will be helpful in understanding these concepts.

— Learn_and_Share

Відповіді:

There is a difference between

independent events: $\mathbb P(A \cap B) =\mathbb P(A)\,\mathbb P(B)$ , i.e. $\mathbb P(A \mid B)= \mathbb P(A)$ so knowing one happened gives no information about whether the other happened
mutually disjoint events: $\mathbb P(A \cap B) = 0$ , i.e. $\mathbb P(A \mid B)= 0$ so knowing one happened means the other did not happen

You asked for a picture. This might help:

— Henry
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Is there a reason you wrote "almost" in the second bullet point? Is that one of those "possible with probability zero" things? I would think it is by definition impossible (such as the probability of heads and the probability of tails), then why write "almost certainly" rather than "certainly"? I suppose this is the probabilistic interpretation.

— gerrit

@Barranka I get that, but that doesn't look like what is drawn in the picture on right. The joint probability of a uniformly drawn random number in [0, 1] being both smaller than 0.4 and larger than 0.6 is not only zero, it is also completely impossible. Isn't that what the wide band in the right figure illustrates? Or am I misreading the figure?

— gerrit

@Barranka I could throw the coin so fast that it escapes earth's gravitational pull. I would venture P(HEADS)=0.499..., P(TAILS)=0.499..., 0<P(LAND ON SIDE)<0.000000000001, and 0<P(ESCAPE VELOCITY)<0.0000000000001. Strictly speaking if the probability of an event is zero, then it can not happen.

— emory

I'm no expert, but even after your last comment I agree with @gerrit: Heads and Tails are disjoint. It's possible to get Not heads and not tails, but it's impossible to get heads and tails. Thus knowing heads happened means that tails could not possibly have happened - no "almost" about it. I could be wrong on my terminology, but if so please explain patiently as I'm not the only one missing it

— Chris H

@Braanka Your coin example is a poor one, as presumably landing on a side has non-zero probability, and if you say that it has zero probability, well, now you're pretty much just begging the question.

— Acccumulation

What I understood from your question, is that you might have confused independent events with disjoint events.

disjoint events: Two events are called disjoint or mutually exclusive if they cannot both happen. For instance, if we roll a die, the outcomes 1 and 2 are disjoint since they cannot both occur. On the other hand, the outcomes 1 and “rolling an odd number” are not disjoint since both occur if the outcome of the roll is a 1. The intersect of such events is always 0.

independent events: Two events are independent if knowing the outcome of one provides no useful information about the outcome of the other. For instance, when we roll two dice, the outcome of each is an independent event – knowing the outcome of one roll does not help determining the outcome of the other. Let's build on that example: We roll two dice, a red and a blue. The probability of getting a 1 on the red is given by P(red = 1) = 1/6, and the probability of getting a 1 on the white is given by P(white = 1) = 1/6. It is possible to get their intersect (i.e. both get 1) simply by multiplying them, since they are independent. P(red = 1) x P(white = 1) = 1/6 x 1/6 = 1/36 != 0. In simple words 1/6 of the time the red die is a 1, and 1/6 of those times the white die is 1. To illustrate:

— Umair Rafique
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The confusion of the OP lies on the notions of disjoint events and independent events.

One simple and intuitive description of independence is:

A and B are independent if knowing that A happened gives you no information about whether or not B happened.

Or in other words,

A and B are independent if knowing that A happened does not change the probability that B happened.

If A and B are disjoint then knowing that A happened is a game changer! Now you would be certain that B did not happen! And so they are not independent.

The only way independence and "disjointedness" in this example are the same is when B is the empty set (which has probability 0). In this case A happening does not inform anything on B

No pictures but at least some intuition

— gota
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