Probability

We know what probability means, but what is its formal definition? Letâ€™s use our intuition to define it. If there is no chance that an event will occur, then its probability of occurring should be 0. On the other extreme, if an event is certain to occur, then its probability of occurring should be 100%, or 1. Hence, our 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:

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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

Whatâ€™s the probability of getting heads twice in a row when flipping a coin twice? Previously we calculated the probability for the first flip to be 1/2. Since the second flip is not affected by the first (these are called 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.

This rule applies to two, three, or any number of consecutive probabilities.