In day to day life, we sometimes make the statements like "It may rain today" or "Train is likely to be late". The words (may, likely, unlikely, chances, doubt) show that the event, we are talking about, is not certain to occur. It may or may not occur.
Probability is a branch of Mathematics which deals with situations involving uncertainty. It originated in the games of chance such as throwing of dice and now probability is used extensively in Biology, Economics, Genetics, Physics, Sociology and other fields.
Random Experiment and Outcome
An experiment in which all possible outcomes are known but the results can not be predicted in advance. A random experiment always has more than one possible outcomes. When the experiment is performed only one outcome out of all possible outcomes comes out. Moreover, we can not predict any particular outcome before the experiment is performed. Repeating the experiment may lead to different outcomes.
Examples
- Tossing of Coin: Suppose we toss a coin. We know in advance that the coin can only land in one of two possible ways - Head (H) or Tail (T).
- Throwing of Die: Suppose we throw a die. We know in advance that the die can only land in any one of six different ways showing up either 1, 2, 3, 4, 5 or 6.
- Drawing a Ball from a bag containing identical balls of different colours without looking into the bag.
- Playing Cards: Drawing a card at random from a well shuffled deck of playing cards. A deck of playing cards consists of 52 cards which are divided into four suits of 13 cards each - spades, hearts, diamonds, and clubs. Spades and clubs are of black colour and others are of red colour. The cards in each suit are ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3, and 2. Cards of kings, queens and jacks are called face cards.
Event
One or more outcomes constitute an event of an experiment. For example, in throwing a die an event could be "the die shows an even number". This event corresponds to three different outcomes 2, 4 or 6.
Event can also be used to describe a single outcome. In case of tossing a coin, “the coin shows up a head” or “the coin shows up a tail” each is an event, the first one corresponds to the outcome H and the other to the outcome T. An event having only one outcome of the experiment is called an elementary event.
Equally Likely Event
Suppose a coin is tossed at random. We have two possible outcomes, Head (H) and Tail (T). We may assume that each outcome H or T is as likely to occur as the other. In other words, the two outcomes H and T are equally likely.
Similarly, when we throw a die, each of the six faces (or each of the outcomes 1, 2, 3, 4, 5, 6) is just as likely as any other to occur. In other words, the six outcomes 1, 2, 3, 4, 5 and 6 are equally likely.
Sample space: Collection of all possible outcomes.
Coin tossed once: S = {H, T}
Coin tossed twice: S = {HH, HT, TH, TT}
Die is thrown once: S = {1, 2, 3, 4, 5, 6}
Sure Event: If number of outcomes favourable to the event is equal to number of total outcomes of the sample space or an event whose probability is 1.
Impossible Event: Having no outcome or an event whose probability is 0.
Probability of Event
The probability of an event E, written as P(E), is defined as:
$$ \text{P(E)} = \frac{\text{Number of outcomes favourable to E}}{\text{Number of all possible outcomes of experiment}} $$
Properties of Probability
- Probability of a sure event is taken as 1.
- Probability of an impossible event is taken as 0.
- P(E) cannot be greater than 1, since numerator being the number of outcomes favourable to E cannot be greater than the denominator (number of all possible outcomes).
- Both the numerator and denominator are natural numbers, so P(E) cannot be negative.
- The sum of the probabilities of all the elementary events of an experiment is 1.
Range of Probability: Probability of an event always lies between 0 and 1 (0 and 1 inclusive).
0 ≤ P(E) ≤ 1
Complementary Event: Event which occurs only when E does not occur.
P(E') = 1 - P(E)
Sum of Probabilities: Sum of all the probabilities is 1.
Examples
Example 1: A coin is tossed once. Find the probability of getting a head.
Let E be the event “getting a head”.
Possible outcomes of the experiment are: Head (H), Tail (T)
Number of possible outcomes = 2
Number of outcomes favourable to E = 1 (i.e., Head only)
$$ P(E) = \frac{1}{2} $$
Example 2: A die is thrown once. What is the probability of getting a number 3?
Let E be the event "getting a number 3".
Possible outcomes of the experiment are: 1, 2, 3, 4, 5, 6.
Number of possible outcomes = 6
Number of outcomes favourable to E = 1 (i.e., 3)
$$ P(E) = \frac{1}{6} $$
Example 3: A die is thrown once. Determine the probability of getting a number other than 3?
Let E be the event “getting a number other than 3” which means “getting a number 1, 2, 4, 5, 6”.
Possible outcomes are: 1, 2, 3, 4, 5, 6
Number of possible outcomes = 6
Number of outcomes favourable to E = 5 (i.e., 1, 2, 4, 5, 6)
$$ P(E) = \frac{5}{6} $$
Example 4: A ball is drawn at random from a bag containing 2 red balls, 3 blue balls and 4 black balls. What is the probability of this ball being of (i) red colour (ii) blue colour (iii) black colour (iv) not blue colour?
Number of possible outcomes of the experiment = 2 (Red) + 3 (Blue) + 4 (Black) = 9
(i) Let E be the event that the drawn ball is of red colour.
Number of outcomes favourable to E = 2
$$ P(E) = \frac{2}{9} $$
(ii) Let E be the event that the ball drawn is of blue colour.
$$ P(E) = \frac{3}{9} = \frac{1}{3} $$
(iii) Let E be the event that the ball drawn is of black colour.
$$ P(E) = \frac{4}{9} $$
(iv) Let E be the event that the ball drawn is not of blue colour. Here “ball of not blue colour” means “ball of red or black colour)
Therefore, number of outcomes favourable = 2 + 4 = 6
$$ P(E) = \frac{6}{9} = \frac{2}{3} $$
Example 5: A card is drawn from a well shuffled deck of 52 playing cards. Find the probability that it is of (i) red colour (ii) face card
Total number of cards = 52
Number of all possible outcomes = 52
(i) Let E be the event that the card drawn is of red colour.
Number of cards of red colour = 13 + 13 = 26 (diamonds and hearts)
So, the number of favourable outcomes to E = 26
$$ P(E) = \frac{26}{52} = \frac{1}{2} $$
(ii) Number of outcomes favourable to the Event E "a face card" = 3 × 4 = 12 (Kings, Queens, and Jacks are face cards)
$$ P(E) = \frac{12}{52} = \frac{3}{13} $$
Example 6: A coin is tossed two times. What is the probability of getting a head each time?
Let us write H for Head and T for Tail.
In this experiment, the possible outcomes will be: HH, HT, TH, TT
So, the number of possible outcomes = 4
Let E be the event “getting head each time”. This means getting head in both the tosses, i.e. HH.
$$ P(E) = \frac{1}{4} $$