**Simple Probability**

#### Exercise 1

What is the probability that a card drawn at random from a deck of cards will be an ace?

##### Solution

Since of the 52 cards in the deck, 4 are aces, the probability is 4/52.

**Conditional Probability**

A conditional probability is the probability of an event
given that another event has occurred. For example, what is
the probability that the total of two dice will be greater
than 8 given that the first die is a 6? This can be computed
by considering only outcomes for which the first die is a
6. Then, determine the proportion of these outcomes that
total more than 8. All the possible outcomes for two dice
are shown in the section on simple probability. There are 6
outcomes for which the first die is a 6, and of these, there
are four that total more than 8 (6,3; 6,4; 6,5; 6,6). The
probability of a total greater than 8 given that the first
die is 6 is therefore 4/6 = 2/3. More formally, this
probability can be written as:

**Probability of A and B**

**Probability of A and B: Independent Variables**

#### Exercise 2

What is the probability that a fair coin will come up with heads twice in a row?

##### Solution

Two events must occur: a head on the first toss and a
head on the second toss. Since the probability of each
event is 1/2, the probability of both events is:

**Probability of A and B: Dependent Variables**

#### Exercise 3

If someone draws a card at random from a deck and then, without replacing the first card, draws a second card, what is the probability that both cards will be aces?

##### Solution

Event A is that the first card is an ace. Since 4 of the
52 cards are aces,

**Probability of A or B**

**Probability of A or B: Mutually Exclusive Variables**

**Probability of A or B:Not Mutually Exclusive Variables**

**Binomial Distribution**

The binomial probability for obtaining

- are dichotomous (fall into only two categories)
- are mutually exclusive
- are independent and
- are randomly selected

#### Exercise 4

Consider this simple application of the binomial distribution: What is the probability of obtaining exactly 3 heads if a fair coin is flipped 6 times?

##### Solution

For this problem,