## Basic Probability, part 5

### August 24, 2009

Let us revise some information.

Pretend we at random choose two cards from a deck at the same time and we want to know what is the probability that we choose a queen or a king. As we know we these two events are mutually exclusive, because one card cannot be and a king and a queen at the same time, so we need to add two probabilities:

**Independent events.**

And now consider the case when we toss a coin an the choose at random on card from a deck. What is the probability that we get Heads and an ace? As we know we need multiple probabilities of those two events: Current two events are called **independent**, because the second event “choose an ace” is not depend on the first event “get Heads”. The probability of getting an ace is and it is independent weather we toss coin or not. The same for coin.

**The multiplication rule 1:**

If event A and event B are independent, then

**Dependent events.**

Let us get a queen from an ordinary deck with 52 cards. The probability of this event is Now put the queen in a pocket. And try to get another queen from the deck. Notice that we have 51 cards and 3 queens! So the second event to get a queen has probability

These two events are called dependent events, the second event is dependent from the first. The probability that event B occurs given that event A has already occurred is denoted as

In our case with two queens the probability that we get a queen in the first choose and it is not replaced, and we get a queen in the second choose is:

**The multiplication rule 2:**

If event A and event B are dependent, then