# 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.

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.

1. p(composite)=p(non-composite)=0.5 hence both are equally likely to occur.
2. 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.

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

 Probability of having cancer or not Test being positive Test being negative cancer 0.05 0.05*0.85=0.00425 0.005*0.15=0.00074 No cancer 0.995 0.995*0.075=0.074625 0.995*0.925=0.920375 Sum 1 0.078875 0.921125

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