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Here's a example with rolling a dice:

$A = \{1, 2\}$
$B = \{2, 3, 4\}$

$P(A \cap B) = P(\{2\})=1/6,$ and $P(A) = 1/3,$ and $P(B) = 1/2,$ so events $A$ and $B$ are independent events.

The definition of independence says, "if event $A$ doesn't influence event $B,$ then these events are independent".

If I add '$1$' to event $B$ and call it event $C,$ i.e., $C = \{1, 2, 3, 4\},$ then $A$ and $C $ are not independent events.

How can I understand this "event $A$ doesn't influence event $B$" more intuitively?

ryang
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user963241
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2 Answers2

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The basic point is that independence of events $A$ and $B$ with positive probability implies $P(B)=P(B \mid A)$ and $P(A)=P(A \mid B)$.

In your example, 1s and 2s are a third of all and also a third of the rolls between 2 and 4 inclusive, so knowing that the roll is between 2 and 4 inclusive doesn't make it any more or less likely that the roll is a 1 or a 2. By contrast knowing that the roll is between 2 and 5 inclusive would make it less likely that it is a 1 or a 2, since 2s are only a quarter of the rolls between 2 and 5.

Ian
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  • I see. That sounds like probability has to be less than or equal to 50% on either side? otherwise, imbalance between probabilities makes the events dependent. – user963241 Dec 14 '20 at 21:19
  • @user963241 Not really, no... – Ian Dec 14 '20 at 23:17
  • So is it not possible to translate this "less likely" in terms of probability percentages? I think it will help if we make it more abstract because the "numbers" 2, 3 4.. don't really matter. All it matters is what probability you have in intersection and probability of A and B. – user963241 Dec 14 '20 at 23:35
  • I am still failing to intuitively see 'independence' thing. In a different question will it be as intuitive as you described or may be different? – user963241 Dec 14 '20 at 23:54
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How can I intuitively see independent events?

This is the question to ask (assuming that event $A$ has a nonzero probability):

  • Does knowing that event A happens change the probability of event $B$ ?

The definition of independence says, "if event $A$ doesn't influence event $B,$ then these events are independent".

What does it mean for event $A$ to influence event $B$? In your given example, if event $A$ happens, then only one of event $B$'s three outcomes are possible, that is, event $B$ becomes restricted to happen in one way, down from three; surely this counts as event $A$ "influencing" event $B$ ? Yet, you have proven from definition that they are independent events.

So, your characterisation of independence of events is wrong or ill-defined.

Here's a example with rolling a dice:

$A = \{1, 2\}$ $B = \{2, 3, 4\}$

These two events $A$ and $B$ are proved to be independent events.

If I add '$1$' to event $B$ and call it event $C,$ i.e., $C = \{1, 2, 3, 4\},$ then $A$ and $C $ are not independent events.

Referring to the boldfaced characterisation above:

  • Knowing that event $A$ happens means that event $B$ can happen only via the outcome '$2$'; but this doesn't change $B$'s probability $\left(\frac12\right),$ since knowing that $A$ happens also reduces the sample space from $\{1,2,3,4,5,6\}$ to $A=\{1, 2\}.$
  • Knowing that event $A$ happens reduces the sample space from $\{1,2,3,4,5,6\}$ to $A=\{1, 2\},$ and at the same time means that event $C$ can happen only via the outcomes '$1$' and '$2$'; that is, knowing that $A$ happens changes $C$'s probability from $\frac23$ to $1.$

Visualising independence.

ryang
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