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Basic probability formulas with examples

Meet with basic probability meaning and some useful formulas for easy problem solving.

Definition of probability

The probability formula is equal to the ratio of the number of favorable outcomes (m) and the total number of outcomes (n).

$$P(A)=\frac{m}{n}$$

The value of probability of an event always must be between 0 and 1, where, value 0 indicates an impossible event and value 1 indicates a certain event.
The opposite probability of an event A is $P(\text{not} \ A) = 1 − P(A)$

What is the probability of rolling a even number on a six-sided die?

Six-sided die has these values: 1, 2, 3, 4, 5, 6. Even number values: 2, 4, 6.

$$P(A)=\frac{3}{6}=\frac{1}{2}$$

Mutually exclusive events

Two events are mutually exclusive when these events can't happen at the same time. The probability of the mutually exclusive events is the sum of their individual probabilities.

$$P(A\ \text{or}\ B)=P(A)+P(B)$$

What is the probability of rolling a 2 or 5 on a six-sided die?

$$P(2\ \text{or}\ 5)=P(2)+P(5)=\frac{1}{6} + \frac{1}{6}=\frac{2}{6}=\frac{1}{3}$$

Not mutually exclusive events

Two events are not mutually exclusive when these events can happen at the same time. The probability of the not mutually exclusive events is the sum of their individual probabilities minus the probability of the two events happening at the same time.

$$P(A\ \text{or}\ B)=P(A)+P(B)-P(A\ \text{and}\ B)$$

What is the probability of rolling a odd or prime number on a six-sided die?

Six-sided die has these values: 1, 2, 3, 4, 5, 6. Odd number values: 1, 3, 5. Prime number values: 2, 3, 5. So values 3 and 5 are included in each event and should only be counted once.

$$P(2\ \text{or}\ 5)=\frac{3}{6} + \frac{3}{6} - \frac{2}{6}=\frac{4}{6}=\frac{2}{3}$$

Independent events

We can calculate the probabilities of two or more independent events by multiplying each probability. Independent events are not affected by each over. For example, probability of A and B equals to the product of probability A and probability B:

$$P(A \ \text{and} \ B) = P(A) × P(B)$$

You flip three coins. What is the probability that you flip three heads?

Solution: P(A and B and C) = P(A) * P(B) * P(C) = 1/2 * 1/2 * 1/2 = 1/8

At the same time you toss a coin and roll a dice. What is the probability of getting a coin head and even number of dice?

The chance to get head is 1/2 and the chance to get even number of dice is also 1/2, because needed numbers are 2, 4, 6 and total number of outcomes is 6, so 3/6 = 1/2.
Now we can get the probability: P(A and B) = 1/2 * 1/2 = 1/4