- 1. Dr. Sonia Bajaj Assistant Professor Department of Zoology Shri Shankaracharya Mahavidyalaya ,Junwani ,Bhilai Probability
- 2. Probability • The term probability is a vague concept, which cannot be mathematically defined. • Probability is the proportion of times an event occurs in a set of trials. • The word 'probability means -chance or possibility. • It is used to predict how likely events are to happen. For example "It may rain today or the local team may win the match' or 'the present P.G. students may fare well in Biostatistics paper, etc. • In each of these statements, there is as much uncertainty as there is certainty. So from the above it follows that probability is subjective and change from person to person. • Probability is calculated by the following formula: P= e/t P=Probability e= number of times an event occurs or frequency t=total number of trials or items • The probability value is always a fraction falling between 0 and 1. 0= means the event is an impossible one and 1= indicates a certain event
- 3. Definition According to Laplace, the French Mathematician "Probability is the ratio of number of favourable cases to the total number of equally likely cases". • If probability is denoted by P. P=Number of favourable cases /Total number of equally likely cases For example, 1. when we toss a coin, there are 2 equally likely results, namely (1) Head and (2) Tail. .. The probability of a head in a toss of a coin is 1 /2 P(H)=1/2 2. When a baby is born, it is male or female. There are two chances for the baby. The male has one chance and the female has another chance. So the probability of male baby is, P= e/t =1/2 = 0.5 Similarly, the probability of female baby is 0.5. When a dice numbered from 1 to 6 is tossed, the total number of chance is 6. The probability of any number is1/6=0.17. Probability study is to reduce the level of uncertainty in decision making. It has therefore special applications in business, administration and research.
- 4. P is the probability of an event to occur. q is the probability of the event not occurring.so p+q=1 When p is known, q can be calculated q=1-p P=1-q When p is 0.17, q is = 1-p =1-0.17 =0.83 Types of Probability 1. Apriori probability 2. Aposteriori probability 1. When the probability is determined before the event takes place, it is called apriori probability or mathematical probability. Example- • Having a male baby • Tossing a head • Getting a number in a dice. 2. When the probability is determined only after the event takes place, it is said to be aposteriori probability or statistical probability. Example- • The number of heads in throwing a coin for 100 times can be calculated only after 100 throws. • The experiment for the calculation of probability is called trial. Eg. Tossing of a coin is a trial. • The results of a trial are called events. Eg. When a coin is tossed, the result will be a head or a tail. Getting a head or a tail is an event. • The occurrence and non-occurrence of a single event is called simple event. Eg. Tossing a single coin.
- 5. Basic Concepts 1.Sample space- The sample space is the set of all possible outcomes. example, when we throw a dice, the possible outcomes are 1, 2, 3, 4, 5 and 6. All these numbers are called sample space. Each possible outcome in a sample space is called sample point. 2. An event An event is said to be a collection of possible outcomes, when an experiment is conducted. example, in the tossing of a coin, the head is an event; the tail is another event. 3. Equi. Probable events When two or more events are equally probable, the events are said to be equi probable events. i.e., when one event has as much chance to occur as the other, they are equi. probable events. They may be also called equally likely events. example, when we toss coin we may get either the head or the tail. Both the events are equally likely or have 50% chance each. 4. Independent events Two events are said to be independent if the occurrence of one does not affect the occurrence of the other. example, when 2 coins are tossed one after another. The result of the first toss does not affect the second toss. Such events are called independent events. 5. Dependent events Two events A and B are said to be dependent if the occurrence of A affects the occurrence of the other B. example, in a pack of cards, there are 52 cards. Suppose. one card is withdrawn, the probability of a King is 4/52 or 1/13. Suppose the card is not replaced, the probability of another King is 3/51 or 1/17. 6. Simple and compound events When a single event takes place the probability of its happening or not happening is known as simple event. When two or more events take place simultaneously, their occurrence is known as compound event.
- 6. Kinds of Probabilities There are two kinds of probability. 1. Mathematical probability 2. Statistical probability In the former, probability is determined before the event has occurred and in the later probability is determined only after the event has occurred. Measures of Probability If an event can happen in 'a' ways and fail to happen in b' ways and all these (a+b) ways are equally likely, the probability of the happening of the event is measured by the ratio a /a+b or b /a+b If the probability of the success of the event is denoted by 'p' and the failure of the event is denoted by ‘q' then
- 8. Theorems of Probability-There are two important theorems of probability- 1. Addition theorem 2. Multiplication theorem 1. Addition theorem-The addition theorem states that 2 events A and B are mutually exclusive, the probability of the occurrence of either A or B is the sum of their individual probability of A and B. (mutually exclusive events).
- 9. 2. Multiplication theorem-This theorem states that if 2 events A and B are independent, the probability that they will both occur is equal to the product of their individual probability. In symbols, if A and B are independent then P (A and B) = P (A) x P (B) llustration-3: A firm has advertised for a steno for its office with three qualities, English knowing, good
- 10. When events are dependent (Conditional Probability) • The multiplication theorem can be applied only when the events are independent. • They cannot be applied when the events are dependent. Two events A and B are said to be dependent when B can occur and when A is known to have occurred or vice versa. The probability of such a type is termed as conditional probability and is denoted by P(A/B). It means probability of A given B has occurred.
- 11. References- 1. Elements of Biostatistics- S. Prasad 2. Biostatistics -Khan & Khanam 3. Basic Concepts of Biostatistics -N. Arumugam