### 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 **multiplication rule**

## Leave a Reply

You must be logged in to post a comment.