Probability - 1

Random experiment

Any action whihc gives one or more results is called a random experiment or trial.Each result of the experiment ic called an outcome of the experiment.

Sample space: The set of all possible outcomes of an experiment is called the sample space of that experiment and is denoted by S.

When the number of possible outcomes are limited or finite it is called finite sample space.

n(s) represents the number of possible outcomes of S.

Example: If we toss a coin once, Head (H) or Tail (T) are possible.

So sample space S is {H,T}

n(S) = 2.

Unbiased experiment
In an unbiased experiment each of the outcomes in the sample space are equally likely to occur.

Event

An event is a subset of a sample space. We can define probabilities for specified events.

An event will be a null set, when the outcomes specified for the event to happen are not in the sample space.

Definition of probability

If S is the finite sample space of an experiment and every outcome of S is equally likely and if E is an event (i.e. E is contained in S), the the probability that E takes place is defined as

P(E) = n(E)/n(S)

Problems on probability

Drawing cards from a pack of cards.

Whenever a number of cards are drawn from a pack of cards, it is understood that, whenever a card is drawn from the pack, it is not replaced in the pack unless otherwise stated in the problem.

1. Two cards are drawn from a well shuffled pack of 52 cards. Find the probability that both are Hearts.

From the pack two cards can drawn in ⁵²C₂ ways.

Therefore n(S) = ⁵²C₂ = 52*51/2

The event two hearts can occur in ¹³C₂ ways as 13 Heart cards are there in the pack. If we identify the event with symbol 2H.

n(2H) = ¹³C₂ = 13*12/2

So P(2H) = n(2H)/n(S) = (52*51)/(13*12) [denominators cancel each other]

= 1/17
The probability of drawing two hearts is 1/17.