Basic Probability
----Understanding Probability
Probability is the quantification of determining the likelihood of an event.
Car manufacturers extend new car customers warranties. These warranties protect the owner in case
something breaks, like non-wear and tear items, such as: transmission, engine, and electrical systems. If
one of these major systems gets damaged while a car is under warranty, the car will be fixed for free.
Car buyers like this protection because it gives them a sense of comfort buying a new automobile.
It is therefore very important that car companies know how long their car parts will last. They need to
know the likelihood of a part failing before a warranty expires. If the part is too likely to fail before the
expiration of the warranty, it will cost the car company money to repair the part and it may also make
the customer feel hesitant to buy a car from the manufacturer.
These probabilities are important in several industries other than the automotive industry. An entire
branch of mathematics, called actuarial science, exists to calculate risk for many industries. Actuaries
study risk for insurance, finance, and government programs involving mortality. retirement, and
predicting the future. Actuaries are respected and well paid.
Probability Calculation
The calculation for probability involves counting events.
The probability of a certain event, called event A, is written as P(A). Probability is then calculated as
follows.
The examples below will run through a variety of circumstances that will demonstrate how to use this
formula.
Spinner Problems
There are many games that use a spinner: like Twister, Life, and Mousetrap. A player spins the arrow on
a spinner, the spinner goes wildly in circles, and then the arrow lands on a space. We will use the
spinner found below to do several examples.
Example 1: Calculate P(spinning 2).
There are eight possible outcomes, {1,2,3,4,5,6,7,8} if we only look at numbers and not colors. The
number two is one of those numbers. So, the probability is...
Example 2: Calculate P(spinning a number < 3).
There are two numbers less than 3, namely {1,2}. There are eight possible outcomes {1,2,3,4,5,6,7,8}.
So, the probability is...
Example 3: Calculate P(an even blue number).
Favorable outcomes will be both blue and even numbers. The blue numbers that are even are {4,6}. So,
there are 2 favorable outcomes out of 8 possible outcomes.
Marble Bag Problems
Probability problems would not be complete if we did not look at a bag of marbles. Marble bag
problems are typically found in math textbooks and they do a great job demonstrating the need to be an
accurate counter. We will use the marble bag below to demonstrate how marble bag problems work.
Example 1: Calculate P(pulling a red marble).
There are three red marbles. There are a total of ten marbles in the bag. The probability is therefore...
Example 2: Calculate P(pulling a green marble).
There are four green marbles. There are a total of ten marbles in the bag. The probability is therefore...
Example 3: Calculate P(pulling a yellow, blue, or red marble).
There is 1 yellow, 2 blue, and 3 red marbles. We have 6 marbles that can be pulled that are considered
favorable outcomes. There are a total of ten marbles in the bag. The probability is therefore...
Dice Problems
Most board games include dice, which means most people have seen 6-sided dice, like these.
When we roll a 6-sided die, the possible outcomes are {1,2,3,4,5,6}. Calculating probabilities with a
single die are relatively simple because they are similar to spinner problems. However, let us look at
what happens when we roll two dice and record the outcomes. Let this table reveal the possible
outcomes.
1
2
3
4
5
6
1
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
2
(2,1)
(2,2)
(2,3)
(2,4)
(2,5)
(2,6)
3
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)
4
(4,1)
(4,2)
(4,3)
(4,4)
(4,5)
(4,6)
5
(5,1)
(5,2)
(5,3)
(5,4)
(5,5)
(5,6)
6
(6,1)
(6,2)
(6,3)
(6,4)
(6,5)
(6,6)
We will use this table to determine the probability problems below.
Example 1: Calculate P(rolling doubles).
There are six cases when doubles are rolled, which are highlighted below.
1
2
3
4
5
6
1
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
2
(2,1)
(2,2)
(2,3)
(2,4)
(2,5)
(2,6)
3
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)
4
(4,1)
(4,2)
(4,3)
(4,4)
(4,5)
(4,6)
5
(5,1)
(5,2)
(5,3)
(5,4)
(5,5)
(5,6)
6
(6,1)
(6,2)
(6,3)
(6,4)
(6,5)
(6,6)
These are the cases as a list: {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}. There are a total of 36 possible
outcomes. So, the probability is...
Example 2: Calculate P(not rolling any 5s).
If we cannot roll any fives, let's look at the cases that involve fives, highlighted below.
1
2
3
4
5
6
1
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
2
(2,1)
(2,2)
(2,3)
(2,4)
(2,5)
(2,6)
3
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)
4
(4,1)
(4,2)
(4,3)
(4,4)
(4,5)
(4,6)
5
(5,1)
(5,2)
(5,3)
(5,4)
(5,5)
(5,6)
6
(6,1)
(6,2)
(6,3)
(6,4)
(6,5)
(6,6)
Counting them reveals eleven cases. Keeping in mind there are a total of 36 possible outcomes and we
cannot consider these highlighted 11 cases, there are therefore 36 - 11, or 25 favorable outcomes. This
means our probability is...
Example 3: Calculate P(rolling a sum < 6).
We have to re-examine the die-rolling table so that it reflects dice sums, as follows.
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
Let's see which sums are less than six.
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
There are ten highlighted sums found above. So, that means our probability is...