Theoretical Probability

Definition of Theoretical Probability

Probability is a likelihood that an event will happen.

We can find the theoretical probability of an event using the following ratio:

Let’s do a couple of examples.

Solved Examples on Theoretical Probability

Example 1

If we toss a fair coin, what is the probability that a tail will show up?

Solution:

Tossing a tail is the favorable outcome here.

When you toss a coin there are only 2 possible outcomes: a Head or a Tail

So the options for tossing a tail are 1 out of 2.

We can also represent probability as a decimal or as a percent.

Example 2

A bag contains 20 marbles. 15 of them are red and 5 of them are blue in color. Find the probability of picking a red marble.

Let’s first answer a few questions here:

If I am going to randomly pick a marble from the bag then what results can I have:

I’ll either pick a red marble or a blue one.

My next question is what the chances of picking a red marble are:

There are 15 red marbles and just 5 blue marbles.

It’s obvious that we have three times as many red marbles as blue marbles.

So, the chance of picking a red marble is more than that of the blue one.

Therefore, the probability of picking a red marble is:

Example 3

Find the probability of getting a sum of 7 when you roll two dice.

Two dice are being rolled. The possible outcomes are as follows:

Let’s use the representation (a, b) for the outcomes where a = number on dice 1 and b = number on dice 2.

(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

There are 36 possible outcomes in all.

The question is when you roll two dice, what are the chances of getting a sum of 7?

From the list above identify the pairs with outcomes that add up to 7.

Let’s highlight them this way:

(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Observe that the pairs along the main diagonal add up to 7. There are 6 such pairs.

So, the probability of getting a sum of 7 when we roll two dice is: