Classical Approach to Probability

Random Experiment and Events: Random experiment is defined as an experiment in which when repeated under essentially identical conditions does not give unique results but may result in any one of the several possible outcomes. These outcomes are known as events or cases. Events are denoted as A, B, C, etc. For example

1. Getting a head (H) or a tail is an event when we loss a coin
2. Getting any of six faces: 1, 2, 3, 4, 5, 6 is an event when we throw a die.
3. Getting an ace or a king or a queen is an event when we draw a card from a pack of well-shuffled cards.

Exhaustive events: The total number of possible outcomes in a random experiment (or trial) are known as exhaustive events. For example:

1. There are two exhaustive events head (H) and tail(T) when tossing a coin.
2. There are six exhaustive events, 1, 2, 3, 4, 5, 6 when we throw a die.

Favourable events: The events which cause the happening of a particular event A, are called the favourable events to the event A. For example:

1. There are three favourable events for the occurrence of an even number (or an odd number) in the throwing of a die.
2. When we draw a card from a pack of cards, there are four favourable events for drawing an ace; there are 12 favourable events fo the drawing of a face card (King, queen, jack).

Mutually exclusive events: Such events where the occurrence of one rules out the occurrence of the other, are called mutually exclusive events. For example:

1. In a tossing a coin there are two mutually exclusive events, for if the head comes in a trial, then tail cannot come in the same train or vice versa.
2. There are 52 mutually exclusive events in drawing a card from a pack of cards.

Equally likely events: The events are said to be equally likely if none of them is expected to occur in preference to other. For example:

1. In tossing a coin, there are two equally likely events H and T.
2. In tossing a die, there are six equally likely events 1, 2, 3, 4, 5, 6.

Classical Definition of Probability: If there are $n$ exhaustive, mutually exclusive and equally likely events, out of which $m$ are favourable to the happening of an event A. Then the probability of happening of A, denoted by P[A], is defined as 

$$P[A]=\frac{Favourable~number~of~case}{Exhaustive ~of ~cases}=\frac{m}{n}$$

Example 1. Find the probability of getting an even number in a throw of a single die.

Solution: Total number of exhaustive cases $n=6$

Total number of favourable cases $m=3$

Required probability $=\frac{m}{n}=\frac{3}{6}=\frac{1}{2}$

Example 2. From a pack of 52 cards, two cards are drawn at random. Find the chance that one is a king and the other a queen.

Solution: Total number of cases is $n=52C_2$

Since there are 4 kings and 4 queens, so the number of favourable cases is $m = 4C_1 \times 4C_1$ 

Required Probability $=\frac{m}{n}=\frac{52C_2 \times 52C_2}{52C_2}=\frac{8}{663}$

Exercises:

1. In a single throw with two dice. Find the probability of getting a total of 10.
2. Two cards are drawn at random from a well-shuffled pack of 52 cards. Find the probability of getting two aces.
3. If $n$ biscuit be distributed among $N$ beggars. Find chance that a particular beggar received $r( > n) $ biscuits.
4. What is the chance that a year selected at random will cointains 53 saturday? 

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