# Probability Addition Theorem Probability of At most, At least, Neither, All One or More Events

# Addition Theorem of Probability

at least ⇔ Union

## Two Non-Disjoint (Non-Mutually Exclusive) Events

For two events A and B are not disjoint (or not mutually exclusive),Probability that at least one of the events would occur

⇒ P(A ∪ B)

= P(A) + P(B) − P(A ∩ B)

## Three Non-Disjoint (Non-Mutually Exclusive) Events

For three events A, B and C which are not disjoint (or not mutually exclusive),Probability that at least one of the events would occur

⇒ P(A ∪ B ∪ C)

= P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C)

# Addition Theorem of Probability and Exhaustive Events

**Experiment**

Drawing a ball from a bag containing balls marked from 1 to 100

Where

S : Set of all elementary events in relation to the experiment

**S** = {1, 2, 3, 4, ... , 99, 100}

The union of sets of exhaustive events is the sample space i.e. the set of all elementary events in relation to the experiment.

Where

E = {2, 4, 6, 8, ... , 98, 100}

O = {1, 3, 5, 7, ... , 97, 99}

F = {5, 10, ... , 95, 100}

T = {3, 6, 9, 12, ... , 96, 99}

Events E and O together form exhaustive events

E ∪ O | = | {1, 2, 3, 4, ... , 99, 100} |

= | S |

Any events combined together with the exhaustive events (E and O here) would also form exhaustive events,

Since E and O together are exhaustive, any three or more events which include E and O would also form exhaustive events.

- E, O and F are exhaustive events
⇒ E ∪ O ∪ F = S.

- E, O and T are exhaustive events
⇒ E ∪ O ∪ T = S

- E, O, F and T are exhaustive events
⇒ E ∪ O ∪ F ∪ T = S

## Sample Space - Probability of occurrence

Set of all elementary events (sample points) in relation to an experiment is the sample space. One or more of the elementary events (or sample points) occur on every iteration of the experiment.P(S) | = |
| ||

= | 1 |

⇒ Probability of occurrence of the sample space is a certainty.

## exhaustive events - Probability

Since the union of exhaustive events is equal to the sample space, the probability of occurrence of the union of (at least one of the) exhaustive events is the same as the probability of the sample space i.e. 1.### Two Exhaustive Events

For two events A and B which are exhaustive, the probability that at least one of the events would occur i.e. the probability of the occurrence of the union of the events is a certainty.P(A ∪ B) = P(S) = 1

### Three Exhaustive Events

For three events A, B and C which are exhaustive, the probability that at least one of the events would occur i.e. the probability of the occurrence of the union of the events is a certainty.P(A ∪ B ∪ C) = P(S) = 1

# Addition Theorem of Probability and Mutually Exclusive Events

## Mutually Exclusive

Two or more events are said to be mutually exclusive if the occurrence of one prevents the occurrence of the others. This happens when there are no sample points or elementary events common to the events### Intersection of Mutually Exclusive Events

Since there are no common elements in the sets representing the events, the set representing the intersection of the events would be a null set.### Mutually Exclusive Events - Probability of occurrence

The probability of occurrence of the intersection of mutually exclusive events is Nil.### Examples

#### Two Events

WhereA = {1, 3, 5, .... (2n − 1) }

B = {2, 4, 6, .... (2n) }

A ∩ B = {} or Φ

⇒ n(A ∩ B) = 0

Events A and B are mutually exclusive.

P(A ∩ B) = n(A ∩ B) n(S) = 0 n(S) = 0 #### Three Events

WhereP = {a, b, c, .... m}

Q = {n, o, p, .... v}

R = {w, x, y, z}

P ∩ Q ∩ R = {} or Φ

⇒ n(P ∩ Q ∩ R) = 0

Events P, Q and R are mutually exclusive.

P(P ∩ Q ∩ R) = n(P ∩ Q ∩ R) n(S) = 0 n(S) = 0

## Mutually Exclusive ⇒ Pair wise Exclusive

Where three or more events are in consideration, the events would be mutually exclusive if and only if they are pairwise mutually exclusive.Events A, B and C are mutually exclusive

only if

Events A and B are mutually exclusive

Events A and C are mutually exclusive

Events B and C are mutually exclusive

## Probability of union of Disjoint (Mutually Exclusive) Events

The probability of occurrence of the union of (at least one of) two or more disjoint (mutually exclusive) events is given by the sum of the probabilities of the individual events.### Two Events

For two events A and B which are disjoint (mutually exclusive),

P(A ∪ B)

= P(A) + P(B) − P(A ∩ B)

= P(A) + P(B) − 0

= P(A) + P(B)

### Three Events

For three events E, F and G which are disjoint (mutually exclusive),

P(E ∪ F ∪ G)

= P(E) + P(F) + P(G) − P(E ∩ F) − P(E ∩ G) − P(F ∩ G) + P(E ∩ F ∩ G)

= P(E) + P(F) + P(G) − 0 − 0 − 0 + 0

= P(E) + P(F) + P(G)

# Addition Theorem of Probability - Mutually Exclusive and Exhaustive Events

## Two Events

For two events A and B which are mutually exclusive and exhaustive,P(A ∪ B) = P(A) + P(B)

Since they are mutually exclusive

P(A ∪ B) = 1

Since they are exhaustive

⇒ P(A ∪ B) = P(A) + P(B) = 1

## Three Events

For three events A, B and C which are mutually exclusive and exhaustive,P(A ∪ B ∪ C) = P(A) + P(B) + P(C)

Since they are mutually exclusive

P(A ∪ B ∪ C) = 1

Since they are exhaustive

⇒ P(A ∪ B ∪ C) = P(A) + P(B) + P(C) = 1

# Probability - Additional relations from set theory

## Two Events

For any two events A and B P(A) = P(A ∩ B) + P(A ∩ B^{c})

P(B) = P(A ∩ B) + P(A^{c} ∩ B)

P(A ∪ B)^{c} = P(A^{c} ∩ B^{c})

P(A ∩ B)^{c} = P(A^{c} ∪ B^{c})

## Three Events

For any three events A, B and C P(A ∪ B ∪ C)^{c} = P(A^{c} ∩ B^{c} ∩ C^{c})

P(A ∩ B ∩ C)^{c} = P(A^{c} ∪ B^{c} ∪ C^{c})

# Occurrence of At least, At most, Only, None of the two or more events

## At least

For two or more events occurrence of at least one of the events implies the occurrence of the event representing the union of those eventsat least ⇔ Union

At least one of the events A and B

⇒ The event A ∪ B

Additionally

A ∪ B | = | 1 − (A ∪ B)^{c} |

= | 1 − (A^{c} ∩ B^{c}) |

At least one of the events P, Q and R

⇒ The event P ∪ Q ∪ R

Additionally

P ∪ Q ∪ R | = | 1 − (P ∪ Q ∪ R)^{c} |

= | 1 − (P^{c} ∩ Q^{c} ∩ R^{c}) |

## At most

### Two Events

For two or more events occurrence of at most one less than the number of events implies the non occurrence (complimentary) of the event representing the intersection of those events.**At most one** of the events A and B

⇒ The event (A ∩ B)^{c}

Non occurrence (complimentary) of the event (A ∩ B)

Additionally

(A ∩ B)^{c} = A^{c} ∪ B^{c}

### Three Events

**At most two** of the events P, Q and R

⇒ The event (P ∩ Q ∩ R)^{c}

Non occurrence (complimentary) of the event (P ∩ Q ∩ R)

Additionally

(P ∩ Q ∩ R)^{c} = P^{c} ∪ Q^{c} ∪ R^{c}

**At most one** of the events P, Q and R

⇏ The event (P ∩ Q ∩ R)^{c}

⇒ P(P ∩ Q^{c} ∩ R^{c}) + P(P^{c} ∩ Q ∩ R^{c}) + P(P^{c} ∩ Q^{c} ∩ R) + P(P^{c} ∩ Q^{c} ∩ R^{c})

## Only

The use of the word only indicates the condition where only the mentioned event should occur and all other events should not occur.### Two Events

**Only one** of the events A and B

⇒ P(A ∩ B^{c}) + P(A^{c} ∩ B)

### Three Events

**Only one** of the events P, Q and R

⇒ P(P ∩ Q^{c} ∩ R^{c}) + P(P^{c} ∩ Q ∩ R^{c}) + P(P^{c} ∩ Q^{c} ∩ R)

**Only two** of the events P, Q and R

⇒ P(P ∩ Q ∩ R^{c}) + P(P ∩ Q^{c} ∩ R) + P(P^{c} ∩ Q ∩ R)