Independence 10/14 and 10/16

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

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

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

Are the two following events independent?

E

The first card is a heart.

F

The second card is a queen.

From the definition of conditional probability, we need to find P(F|E) by computing

$$P(E\cap F)=P(1st\; is\; \heartsuit \; and\; 2nd\; is\; Q )=P(\heartsuit,Q)=\frac{51}{51 \times 52}$$

and $P(E)=\frac{1}{4}$.

or we showed that equivalently, E and F are independent if and only if:

$$P(E\cap F)=P(E)\times P(F)$$

Beware this multiplication rule is ONLY available if and only if the events ARE independent.

Example:
De Méré's problem is whether or not it is more likely to get at least one double six in 24 throws of a pair of dice or to get at least one six in 4 throws of a die?

Essential to this argument is the fact that each throw of a die is independent of the preceding one.

The easier probability to compute is the complementary one:

$$P(no\; double\; sixes\; in\; twenty\; four\; throws)= (\frac{35}{36})^{24}=.509$$

and the complementary event

$$P(at\; least\; one\;double\;six)=.491$$

$$P(at \; least\; one\; six\; in\; four\; throws)=1-P(no \; sixes\; in\; 4 \;throws)= 1-(\frac{5}{6})^{4}=1-.482=.518$$