Probability Formula

Probability Formula
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Random Experiments and Sample Space

Randon Experiments: An experiment is called random if it results in two or more outcomes and it is not possible to tell (predict) the outcome in advance.

Sample Space: The collection of all possible outcomes of a random experiment is called sample space.

Sample Space of throwing a die

Sample Space (S) = {1, 2, 3, 4, 5, 6}

Cardinal Number n(S) = 6

Events

A subset of the sample space associated with a random experiment is called an event.

Types of event

  1. Compound event:
  2. Sure event:
  3. Impossible event :
  4. Equally likely outcomes:

Compound Event: A compound event is an event that consists of two or more simple events, also known as elementary events. These events are considered together as a single event.

For example, if you are rolling a six-sided die and you want to know the probability of getting either a 4 or a 5, the event “rolling a 4 or a 5” is a compound event.

Sure Event (Certain Event): A sure event, also known as a certain event, is an event that is guaranteed to happen. Its probability is 1, which means it will occur with certainty.

For example, if you toss a fair coin, the event “getting either heads or tails” is a sure event because one of these outcomes is guaranteed.

Impossible Event: An impossible event is an event that cannot happen. Its probability is 0, which means it will not occur under any circumstances.

For example, if you roll a six-sided die and want to know the probability of getting a 7, this is an impossible event because the die only has numbers from 1 to 6.

Equally Likely Outcomes: Equally likely outcomes refer to a situation where all possible outcomes of an event have the same probability of occurring.

For instance, when rolling a fair six-sided die, each of the six faces (1, 2, 3, 4, 5, and 6) has an equal probability of 1/6 of being rolled, assuming the die is fair. In this case, the outcomes are equally likely.

Algebra of events

  1. Complement of an event: The complement of an event is the set of all outcomes not belonging to that event. Denote – E’ or Ec
  2. The event A or B (A ∪ B): The event A or B (A ∪ B) represents the union of events A and B, including all outcomes that are in either A, B, or both.
  3. The event A and B (A ∩ B): Event A and B (A ∩ B) represents the intersection of events A and B, consisting of outcomes that are common to both events.
  4. The event A but not B (A – B): “A but not B” means outcomes that are in event A but not in event B.  Also A ∩ B’

Mutually exclusive events

Two or more events associated with a random experiment are called mutually exclusive if the occurrence of any one of them excludes the occurrence of the other events.

i.e. (A ∩ B) = Ø

Exhaustive events

The events A and B associated with a random experiment with sample space S are called exhaustive events if (A ∪ B) = S

Mutually exclusive and exhaustive events

Let S be the sample space associated with a random experiment, then the events A and B of S are called Mutually exclusive and exhaustive if

  1. (A ∩ B) = Ø
  2. (A ∪ B) = S

Probability

Probability is the measure of uncertainty.

P(A) = Number\ of\ outcomes\ favourable\ to\ E\over Total\ number\ of\ possible\ outcomes

Note:

  1. 0 ≤ P(A) ≤ 1
  2. P (A) + P(A’) = 1 or P (A) + P(not A) = 1
  3. P (Sure event) = 1
  4. P (Impossible event)  = 0

Odd in favour and odds against

Odd in favour = Number\ of\ favourable\ outcomes\over Number\ of\ unfavourable\ outcomes

Odds against = Number\ of\ unfavourable\ outcomes \over Number\ of\ favourable

Laws of Probability

Theorem 1

If A and B are events associated with a random experiment having Sample Space S and if A ⊂ B, then

  1. P (A) ≤ P(B)
  2. P(B – A) = P(B) – P(A)

Theorem 2

If A and B are events associated with a random experiment having Sample Space S, then

  1. P (A ∪ B) = P(A) + P(B) – P(A ∩ B)
  2. P (A ∪ B) = P(A) + P(B) [For mutually exclusive]
  3. P (A ∪ B) = P(A) + P(B) = 1 [For mutually exclusive and exhaustive]
  4. P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(A ∩ C) + P(A ∩ B ∩ C)
  5. P(A ∪ B ∪ C) = P(A) + P(B) + P(C) [For mutually exclusive]
  6. P(A ∪ B ∪ C) = P(A) + P(B) + P(C) = 1 [For mutually exclusive and exhaustive]

Theorem 3

If A and B are events associated with a random experiment having Sample Space S, then

  1. P(A ∪ B) = P(A – B) + P (B – A) + P (A ∩ B)
  2. P(A ∪ B) = P(A) + P(B – A) = P(B) + P(A – B)
  3. P(A) = P (A – B) + P (A ∩ B)
  4. P(B) = P (B – A) + P (A ∩ B)
  5. P(A – B) = P(A ∩ B’)
  6. P (B – A) = P(A’ ∩ B)
  7. P(A) + P(B) = P(A – B) + P (B – A) + 2P (A ∩ B)

Conditional Probability

If A and B are two events associated with the same sample space of a random experiment, the conditional probability of the event A given that B has occurred, i.e. P (A|B) is given by

P (A|B) = P(A∩B)\over P(B)

Also, the conditional probability of the event BA given that A has occurred, i.e. P (B|A) is given by

P (B|A) = P(A∩B)\over P(A)

Properties of conditional probability

Let A and B be events of a sample space S of an experiment, then we have

  1. P(S|B) = P(B|B) = 1
  2. P(A′|B) = 1 − P(A|B)

Multiplication Theorem on Probability

  1. P(A ∩ B) = P(A) P(B|A)
  2. P(A ∩ B) = P(B) P(A|B)

provided P(A) ≠ 0 and P(B) ≠ 0

Independent Events

Two events E and F are said to be independent, if

  1. P(B|A) = P (B) provided P (A) ≠ 0
  2. P (A|B) = P (A) provided P (B) ≠ 0

Let A and B be two events associated with the same random experiment, then A and B are said to be independent if

P(A ∩ B) = P(A) . P (B)

Remark:

Three events A, B and C are said to be mutually independent, if

  1. P(A ∩ B) = P(A) P(B)
  2. P(A ∩ C) = P(A) P(C)
  3. P (B ∩ C) = P(B) P(C)
  4. P(A ∩ B ∩ C) = P(A) P (B) P(C)

Theorem of total probability

Let {E1 , E2 ,…,En } be a partition of the sample space S, and suppose that each of the events E1 , E2 ,…, En has a nonzero probability of occurrence. Let A be any event associated with S, then

P(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + … + P(En) P(A|En)

or, P(A) = \sum_{i=1}^{n}{P(E_i)P({A| E_i})}

Bayes’ Theorem

If E1, E2,…, En are n non-empty events which constitute a partition of sample space S, i.e. E1, E2, …, En are pairwise disjoint and E1∪ E2∪ … ∪ En = S and A is any event of nonzero probability, then

P(Ei|A) = P(A|E_i)\over \sum_{i=1}^{n}{P(E_i)P({A| E_i})}

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