# Probability

*probability*should be a number between 0 and 1, inclusive. But what kind of number? Suppose your favorite actor has a 1 in 3 chance of winning the Oscar for best actor. This can be measured by forming the fraction 1/3. Hence, a

*probability*is a fraction where the top is the number of ways an event can occur and the bottom is the total number of possible events:

**Example:**

*Flipping a coin**Whatâ€™s the probability of getting heads when flipping a coin?*

There is only one way to get heads in a coin toss.

Hence, the top of the probability fraction is 1.

There are two possible results: heads or tails.

Forming the probability fraction gives 1/2.

**Example:**

*Tossing a die**Whatâ€™s the probability of getting a 3 when tossing a die?*

A die (a cube) has six faces, numbered 1 through 6.

There is only one way to get a 3.

Hence, the top of the fraction is 1.

There are 6 possible results: 1, 2, 3, 4, 5, and 6.

Forming the probability fraction gives 1/6.

**Example:**

*Drawing a card from a deck**Whatâ€™s the probability of getting a king when drawing a card from a deck of cards?*

A deck of cards has four kings, so there are 4 ways to get a king.

Hence, the top of the fraction is 4.

There are 52 total cards in a deck.

Forming the probability fraction gives 4/52, which reduces to 1/13.

Hence, there is 1 chance in 13 of getting a king.

**Example:**

*Drawing marbles from a bowl**Whatâ€™s the probability of drawing a blue marble from a bowl containing 4 red marbles, 5 blue marbles, and 5 green marbles?*

There are five ways of drawing a blue marble.

Hence, the top of the fraction is 5.

There are 14 (= 4 + 5 + 5) possible results.

Forming the probability fraction gives 5/14.

**Example:**

*Drawing marbles from a bowl (second drawing)**Whatâ€™s the probability of drawing a red marble from the same bowl, given that the first marble drawn was blue and was not placed back in the bowl?*

There are four ways of drawing a red marble.

Hence, the top of the fraction is 4.

Since the blue marble from the first drawing was not replaced, there are only 4 blue marbles remaining.

Hence, there are 13 (= 4 + 4 + 5) possible results.

Forming the probability fraction gives 4/13.

# Consecutive Probabilities

*independent*events), its probability is also 1/2. Forming the product yields the probability of two heads in a row: .

Whatâ€™s the probability of drawing a blue marble and then a red marble from a bowl containing 4 red marbles, 5 blue marbles, and 5 green marbles? (Assume that the marbles are not replaced after being selected.) As calculated before, there is a 5/14 likelihood of selecting a blue marble first and a 4/13 likelihood of selecting a red marble second. Forming the product yields the probability of a red marble immediately followed by a blue marble: .

These two examples can be generalized into the following rule for calculating consecutive probabilities:

**Note:**To calculate consecutive probabilities, multiply the individual probabilities.