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### Theoretical Probability

When probability of an event is calculated without actually performing the experiment, it is called theoretical probability.

For example, a number cube has six faces, numbered 1 to 6. Out of these six numbers, three are even (2, 4 and 6). Thus, we can write,
Total number of possible outcomes = 6
Number of favorable outcomes (of even numbers) = 3
Now we write the formula for theoretical probability.
P(A) = "Number of favorable outcomes"/"Total number of possible outcomes"
Put the values of the number of favorable outcomes and total number of outcomes,
P("even number") = 3/6
Simplify the fraction,
P("even number") = 1/2
Thus, the probability of getting an even number on rolling a number cube is 1/2.

In the above problem, we calculated the probability without actually doing an experiment, but with just the concept of the number of favorable events, the total number of events and the formula for theoretical probability.

Thus we see that theoretical probability is simply calculated in three steps:
• Calculate the total number of outcomes in the sample space
• Calculate the number or favorable outcomes in the sample space
• Put these values in the above formula
The following examples will make the concept of theoretical probability clearer.

#### Problem 1

What is the probability of getting tails on tossing a coin?

A coin has two sides, heads and tails. On tossing it, we can get either heads, or tails. Thus, the sample space is {heads, tails} and the total number of possible outcomes is 2. Further, there is only one "tails" in the sample space. Thus, the number of favorable outcomes is 1. Now we apply the formula for theoretical probability,
P(A) = "Number of favorable outcomes"/"Total number of possible outcomes"
That is,
P("tails") = 1/2
Thus, the probability of getting tails on tossing a coin is 1/2.

#### Problem 2

What is the probability of getting the number 3 on rolling a number cube?

As explained in the first example in this post, there are six faces on a number cube numbered 1 to 6. On rolling the cube, you can get any one of the six faces facing up. Thus the total number of possible outcomes is 6. Now only one face of the number cube is numbered 3. Thus the number of favorable outcomes is 1. Thus,
P(A) = "Number of favorable outcomes"/"Total number of possible outcomes"
Putting the values in the formula,
P(3) = 1/6
Thus, the probability of getting the number 3 on rolling a number cube is 1/6.

#### Problem 3

What is the probability of getting a prime number on rolling a number cube?

There are four prime numbers on a number cube: 1, 2, 3 and 5. Total number of numbers on it are 6. Thus, the number of favorable outcomes is 4 while the total number of possible outcomes is 6. Thus,
P("prime number") = "Number of prime numbers"/"Total number of numbers"
P("prime number") = 4/6 = 2/3
Thus, the probability of getting a prime number on rolling a number cube is 2/3.

#### Problem 4

What is the probability of selecting a red card from a standard deck of 52 cards?

There are 26 red and 26 black cards in a deck of 52 playing cards. On picking one card, we can get any one of the 52 cards. Thus, the total number of possible outcomes is 52. Since there are 26 red cards, the picking any one of the 26 red cards will result in a favorable outcome. Thus, the number of favorable outcomes is 26.
Applying the formula for theoretical probability,
P(A) = "Number of favorable outcomes"/"Total number of outcomes"
P("red card") = 26/52 = 1/2
Thus, the probability of getting a red card when picking one card out of a deck of 52 cards is 1/2.

#### Problem 5

There are four red, three yellow and five blue marbles in a bag. What is the probability that you pick a blue marble from the bag?

Total number of marbles in the bag is 4 red + 3 yellow + 5 blue = 12. Picking a marble can result in any one of the 12 marbles, thus the total number of possible outcomes is 12. Further, picking any one of the five blue marbles will result in a favorable outcome. Thus, the number of favorable outcomes is 5. Applying the formula for theoretical probability,
P(A) = "Number of favorable outcomes"/"Total number of outcomes"
P("blue marble") = 5/12