Probability calculator

Basic Rules for Finding Probabilities

The addition rule

is used to calculate the probability that event A or event B occurs. To apply this rule, add the probabilities for events A and B, then subtract the probability of intersection. So for the union of two events, we have the following formula:

P(A or B) = P(A) + P(B) - P(A and B)

Example 1:

Consider families with two children. Let A be the event that the first child is a girl, and B be the event that the second child is a girl. In this case, P(A and B) is the probability that both children are girls, and P(A or B) is the probability that at least one child is a girl. We know that P(A) = 1/2, P(B) = 1/2, and P(A and B) = 1/4, then:

P(A or B) = P(A) + P(B) - P(A and B) =
= 1/2 + 1/2 - 1/4 = 3/4

The multiplication rule

is used to find the probability that events A and B both occur. For independent events, multiplication rule is

P(A and B) = P(A) × P(B)

and for dependant events, the formula is:

P(A and B) = P(A) × P(B|A).

Example 2:

Suppose we flip the coin three times. What is the probability of getting all three heads?

Let A be the event that we get a head on a single coin toss. The P(A)=1/2. The P(AAA) is:

P(AAA) = P(A) × P(A) × P(A) =
= 1/2 × 1/2 × 1/2 = 1/8

Binomial distribution

A binomial experiment involves repeated trials with each trial having two possible outcomes: success or failure.

A binomial distribution simply counts the number of successes across n independent trials.

The probability of having k successes in n trials is

Example 3:

What is the probability of getting exactly 2 heads in 6 coin tosses?

The number of trials is n=6.

The number of successes is k=2.

The probability od success is p=1/2.

Now, we can apply above formula

P(k) = n!/(k! × (n-k)!) × p^k × (1-p)^n-k
P(2) = 6!/(2! × (6-2)!) × 0.5² × (1-0.5)^6-2
P(2) = 720/(2 × 24) × 0.25 × 0.0625
P(2) = 15 × 0.25 × 0.0625
P(2) = 15 × 0.25 × 0.0625
P(2) = 0.234
P(2) = 23.4%