Let us consider an experiment in the pack of 52 playing cards. Each drawing of a card is called an event.
It will be 52 events if we draw one card from a pack of 52 cards at a time. Those events are designated as E1, E2,….E52.
Second examples of events throwing a dice and tossing a coin. And the outcome of a single result of an experiment is called an event.
Random Experiment: Each trial experiment is performed under identified conditions; the outcome is not always the same, but any possible outcome than the experiment is called a random experiment.
Random experiment as drawing a card from a well-shuffled pack of cards and tossing a coin.
Sample space: The sample (S) is a set of all possible outcomes in a random experiment. Consider the following example;
- Tossing a fair coin, we have,
- Throw of dice, we get,
- When two coins are tossed together, the possible outcome,
- Together two dice are thrown,
Simple event: Each outcome of an experiment is called a simple event.
Event: Any combination of simple events is known as an event. Event denoted as upper case letter E.
For example, when die is thrown, then each one of 1,2,3,4,5,6 is a simple event, and getting a prime number is an event: E ={ 2,3,5}
Mutually Exclusive event: Mutually Exclusive event is known as a set of events. If the happening of one event excludes the happening of the other. So, E1 and E2 are mutually exclusive if,
If E1 Ո E2 =Փ, here Փ = empty set
If E1 Ո E2 =Փ, then E1 and E2 are called compatible events.
Consider throwing a dice; we have,
S = {1,2,3,4,5,6}
Suppose E1 be the event of getting a number less than 3; clearly, E1 = {1,2} E2 be the event of getting a number greater than 4. So, E2 = {5, 6} ; so, E1 Ո E2 =Փ
Exhaustive events: The events E1, E2,….., Ek that E1 Ս E2 Ս….Ս Ek = S, are called Exhaustive events.
Equally likely events: If none of them is expected to occur in performance, the other is said to be equally likely events.
Drawing a card from a pack of well-shuffled cards results in 52 equally likely events.
Probability: Random experiment let S be the sample space and E be a subset of S, i.e. E S. Then, E is an event, so,
odds in favour of an event and odds against it: if m is the number of ways an event can occur and n is the number of ways in which it does not occur, then,
Complementry Event: Suppose S is the sample space and E Then, E is an event. Also, Ec S. So, Ec is an event called complementary event, and E or E denotes it.
Additional Theorem
If E1 Ո E2 = Փ (i.e. an empty set), then
Independent events: if two events are said to be independent, one does not depend on the other. For example, tossed are two coins.
Let E1 be the event of getting a head on the 1st coin and E2 be the event of getting a head on the 2nd coin, not depend on the occurrence of a head on the 1st coin. So, E1 and E2 are independent events.
Multiplication Theorem: If E1 and E2 are independent events, then,
Conditional probability: when an event E2 has already occurred and the probability of the occurrence of an event E1 is called conditional probability, P (E1/E2). It can be shown that,
Binominal Theorem of probability: suppose n independent trials of an experiment with P as the probability of failure. Then,
Mathematical expectation: x is a random varible having values x1, x2, x3,……xn with corresponding probability P1, P2, P3,…..Pn, then the mathematical expectation of x is defined as,
