### Probability Series

### Introduction

**Trail or Experiment**- The act that leads to a result with certain possibility.**Sample Space**- Set of all possible outcomes of an experiment.**Event**- Non empty subset of a sample space.

### Basic Probability Formula

\[P(A) = \sum_{i=1}^n P(E_i) \label{1} \tag{1} \]

Where A is an event, S is the sample space, \(E_1 … E_n\) are the n outcomes in A.

If \(E_1 … E_n\) are equally likely to occur, then \eqref{1} can be written as,

\[P(A) = {\text{number of outcomes in A} \over \text{total number of possible outcomes}} \label{2} \tag{2}\]

From \eqref{2}, following results can be inferred,

- \(0 \leq P(A) \leq 1\),
- \(P(S) = 1\)

### Complement of an Event

Compliment of an event A is defined as all the outcomes of the sample space, S that are not in A, i.e.

\[P(A^c) = 1 - P(A) \tag{3} \]

Where \(A^c\) is used to denote the compliment of A.

### Union and Intersection of Events

\[P(A \cup B) = P(A) + P(B) + P(A \cap B) \label{4} \tag{4}\]

### Mutually Exclusive Events

Two events A and B are mutually exclusive if there are no overlapping outcomes, i.e., the intersection of the two experiments is a null set.

\[P(A \cap B) = 0 \label{5} \tag{5}\]

Using \eqref{4} and \eqref{5},

### Independent Events

Two events A and B are independent if occurence of one does not affect the probability of the other occuring and is mathematically given by,

\[P(A \cap B) = P(A) * P(B)\]

### Sum Rule or Marginal Probability

\[P(A) = \sum_{B} P(\text{A and B})\]

### EXAMPLE

- M wants to go fishing this weekend to nearby lake.
- His neighbour A is also planing to go to the same spot for fishing this weekend.
- The probability that it will rain this weekend is \(p_1\).
- There are two possible ways to reach the fishing spot (bus or train).
- The probability that
- M will take the bus is \(p_{mb}\)
- A will take the bus is \(p_{ab}\).

- Travel plans of both are independent of each other and rain.
- What is the probability \(p_{rs}\) that M and A meet each other only (should not meet in bus or train) on a lake in rain ?

```
p_mb = float(input()) # P(M taking Bus)
p_ab = float(input()) # P(A taking Bus)
p_1 = float(input()) # P(Rain)
p_rs = p_1 * (1 - p_ab*p_mb - (1-p_ab)*(1-p_mb)) # P(Meet at lake only)
print("%.6f" % p_rs)
```