Conditional Probability

Probability vs Statistics

Probability is used to predict the likelihood of a future event.

vs

Statistics is used to analyse past events

Basics of Probability

Probability in simple terms is the likelihood of a situation happening. When unsure of the outcome, the probability can be calculated to know its chances.

Probability(Event)=(Number of favourable outcomes of an event) / (Total Number of possible outcomes)

The most simple example of this is a coin toss:

A coin toss can have one of 2 outcomes either heads or tails and both have an equal chance of happening. So the probability of either situation happening is 0.5.

Mathematically: p(heads)= 0.5 and p(tails)=0.5

The probability of certain outcomes occurring can be uneven as well.

For example: the probability of getting a prime number between 1 to 10.

In 1 to 10, there are 2,3,5,7 i.e. 4 out of 10

Hence the probability becomes 4/10 = 0.4

In 1 to 10, there is one unique number 1 i.e. 1/10

Hence probability becomes 1/10 or 0.1

In 1 to 10 there are5 composite numbers i.e. 4,6,8,9,10

Hence the probability becomes 5/10= 0.5

Properties of probability:

1. The probability of an event can only be between 0 and 1 and can also be written as a percentage.
2. The probability of event A is often written as P(A).
3. If P(A) > P(B), then event A has a higher chance of occurring than event B.
4. If P(A) = P(B), then events A and B are equally likely to occur.
5. The Sum of probabilities of all possible outcomes of a situation is always 1.
6. The probability of a sure situation is 1
7. The probability of an impossible event is 0

Continuing with the above example it is seen that

p(prime)= 0.4

p(unique)=0.1

p(composite)=0.5

1. All probabilities are between 0 and 1
2. p(composite)> p(prime) > p(unique)

Shows that selecting a composite has a higher chance of occurring than selecting a prime or unique number.

4. p(composite)=p(non-composite)=0.5 hence both are equally likely to occur.
5. Sum of probabilities

p(prime) + p(composite) + p(unique) = 0.4 + 0.1 + 0.5

= 1

If A and B are two independent events, then the probability that both will occur is equal to the product of their individual probabilities.

p(A ⋂ B)= p(A)* p(B)

Conditional Probability

Conditional probability is defined as the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome.

This is mathematically denoted by p(A/B), which means the probability of A given B has already occurred. 

p(A/B) = p(A ⋂ B)/ p(B)

70% of your friends like Chocolate, and 35% like Chocolate AND like Strawberries.

What percent of those who like Chocolate also like strawberries?

P(Strawberry|Chocolate) = P(Chocolate and Strawberry) / P(Chocolate)

0.35 / 0.7 = 50%

Bayes theorem

Bayes’s Theorem provides a principled way of calculating conditional probability.

Image source: https://neelshelar.com/wp-content/uploads/2020/08/BT.png

An example of a use-case for the Bayes theorem

The probability of a woman getting cancer in a region is 0.05 The machine used to test this has an 85% chance of being correct, and a woman without cancer has a 92.5% chance of getting a negative result. If a woman comes in to get tested and it comes out positive. What is the probability that she has cancer?

P(Cancer) = 0.005

P(Test Positive | Cancer) = 0.85

P(Test Neg|No cancer) = 0.925

Using the Bayes theorem:

P(Cancer|Test Positive)= P(Cancer) * P(Test Positive | Cancer) / P(Test Positive)

For this P(Test Positive) is still required

Let’s create a probability table for the same

This  is one of the cases when the Bayes theorem is used in real life.