Theoretical Probability

Probability is a likelihood that an event will happen.
We can find the theoretical probability of an event using the following ratio:
P (event) = Number of favorable outcomes / Total number of outcomes
Lets do a couple of examples.

Example 1

Find the probability of tossing a tail.
This means that tossing a tail is the favorable outcome here.
When you toss a coin there are only 2 outcomes: a Head or a Tail
So the options for tossing a tail are 1 out of 2.

P(tail) = number of favorable outcomes / total number of outcomes = 1/2.
We can also represent probability as a decimal or as a percent.
So, P (tail) = = 0.5 = 50%.


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.

If I am going to pick a marble randomly then what results can I have:
Ill 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.
Its 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 = number of favorable outcomes / total number of outcomes = 15 / 20 = 3 / 4 = 0.75 = 75%.

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:
Lets 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.

Lets 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 = number of favorable outcomes / total number of outcomes = 6 / 36 = 1/6 approx. 0.17 approx. 16.67%.

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