Independence of events: intuitive definitions matter
First and foremost: independence of an AP Statistics course from Math is nonsense. Most of Stats is based on mathematical intuition.
Independent events
The usual definition says: events are called independent if
(1)
You can use it formally or you can try to find a tangible interpretation of this definition, which I did. In Figure 1, the sample space is the unit square. Let
In Mathematica, enter the command
Animate[ParametricPlot[{{0.2 + a, t}, {0.4 + a, t}, {t, 0.3 + b}, {t,
0.6 + b}}, {t, 0, 1}, PlotRange -> {{0, 1}, {0, 1}},
PlotRangeClipping -> True, Frame -> True,
PlotStyle -> {Red, Red, Blue, Blue}, Mesh -> False], {a, -0.15,
0.55}, {b, -0.25, 0.35}, AnimationRunning -> False]
Choose "Forward and Backward" and then press both Play buttons. Those who don't have Mathematica, can view my video.
The statement "If A and B are dependent events, then so are A and the complement of B" (Agresti and Franklin, p.237) is not so simple. Here is the formal proof of the complementary statement ("dependent" is replaced with "independent";
Reading equation (1) from left to right: in practice, if we know that events are independent, we can find the probability of the joint event
Reading equation (1) from right to left: in theory, if we want our events to be independent, we can define the probability of the joint event
Why there is division in the definition of conditional probability?
Golovkin crushed Brook, and I am happy. Let
To satisfy the completeness axiom, we divide both sides by
This explains why conditional probabilities are defined by
(2)
If
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