Independence

Intuitive definition:
Knowing that another event B occurs does not affect the probability of the event A of occurring, this means that A and B are independent.

As a formula: $P(A\;\; given\;\;B)=P(A\vert B)=P(A)$

Mistake in class
Mistakenly I wrote too quickly at the end of lecture an expression that was wrong, the right expression is:

$$P(A\; and\; B)= P(A) \times P(B\; given\; A)$$

The only inequality that is always true is

$$P(A \; and\; B) \leq P(A)$$

Example of how the conditional probabilities can both be bigger or smaller:
In considering colorblindedness, suppose I consider the binary random variables associated to color blindness and gender (associate 0 if male, 1 if female), these are called indicator variables, we can tabulate the probabilities of all 4 possible pairs of outcomes as:

$$\begin{array}{ll\vert ll\vert l} && Male & Female & Total\\ ... ... \hline Total &&\frac{1}{2}&\frac{1}{2}&\\ \hline \end{array}$$

So that from this table of joint distribution we read:

$$\begin{eqnarray*} P(colorblind)&=&P(C)=\frac{18}{512}\\ P(man)&=&P(M)=\frac{1}{... ...\frac{\frac{2}{512}}{\frac{1}{2}}=\frac{2}{256}=\frac{1}{128}\\ \end{eqnarray*}$$

We see that for women: $P(C\vert W)\leq P(C)$
And for men $P(C\vert M)\geq P(C)$

When $A$ and $B$ are not independent
Sometimes we have $P(A \; given\; B) \leq P(A)$,

Sometimes we have $P(A \; given\; B) \geq P(A)$,

When two events are independent the probability of them both occurring is just the product of their probabilities.

The probability of throwing a double three with two dice is the result of throwing three with the first die and three with the second die. The total possibilities are, one from six outcomes for the first event and one from six outcomes for the second, Therefore (1/6) * (1/6) = 1/36.

The two events are independent, since whatever happens to the first die cannot affect the throw of the second, the probabilities are therefore multiplied, and remain 1/36.

Definition:Two events $E$ and $F$ are said to be independent if

$$P(E\vert F)=P(E)=P(E\vert F^c),\qquad P(F)>0$$

Examples:
We draw two cards one at a time from a shuffled deck of $52$ cards.

In class we looked at the foolowing two events:

A

The first card is a 7 $\clubsuit$.

B

The second card is a queen $\heartsuit$.

We saw that

$$P(B\vert A) \neq P(B)$$

These events are not independent.