This document discusses probability and chance. It defines probability as a measure from 0 to 1 of how likely an event is to occur, with 0 being impossible and 1 being certain. Chance is expressed as a percentage to indicate likelihood. Examples are given of using probability to predict weather or outcomes of rolling dice. The history of probability is outlined as emerging from games of chance. Modern uses include traffic control and genetics. Some occurrences can be predicted through science, while others like coin tosses remain unpredictable. Formulae are provided for calculating probabilities from sample data. The document concludes with applications in risk assessment and markets.

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2. Probability and Chance

3. What Is Probability?Probability is a measure of how likely it is for an event to happen. We name a probability with a number from 0 to 1. If an event is certain to happen, then the probability of the event is 1. If an event is certain not to happen, then the probability of the event is 0. We can even express probability in percentage.

4. ChanceChance is how likely it is that something will happen. To state a chance, we use a percent. Certain not to happen ---------------------------0% Equally likely to happen or not to happen ----- 50 % Certain to happen ----------------------------------- 100%

5. Examples Of ChanceWhen a meteorologist states that the chance of rain is 50%, the meteorologist is saying that it is equally likely to rain or not to rain. If the chance of rain rises to 80%, it is more likely to rain. If the chance drops to 20%, then it may rain, but it probably will not rain.

6. Donald is rolling a number cube labeled 1 to 6. Which of the following is LEAST LIKELY? an even number an odd number a number greater than 5

7. The History of Probability Probability originated from the study of games of chance. Tossing a dice or spinning a roulette wheel are examples of deliberate randomization that are similar to random sampling. Games of chance were not studied by mathematicians until the sixteenth and seventeenth centuries. Probability theory as a branch of mathematics arose in the seventeenth century when French gamblers asked Blaise Pascal and Pierre de Fermat (both well known pioneers in mathematics) for help in their gambling. In the eighteenth and nineteenth centuries, careful measurements in astronomy and surveying led to further advances in probability.

8. Modern Use Of ProbabilityIn the twentieth century probability is used to control the flow of traffic through a highway system, a telephone interchange, or a computer processor; find the genetic makeup of individuals or populations; figure out the energy states of subatomic particles; Estimate the spread of rumors; and predict the rate of return in risky investments.

9. Predictable and Unpredictable OccurrencePredictable Occurrences:The time an object takes to hit the ground from a certain height can easily be predicted using simple physics. The position of asteroids in three years from now can also be predicted using advanced technology.Unpredictable Occurrences:Not everything in life, however, can be predicted using science and technology. For example, a toss of a coin may result in either a head or a tail. Also, the sex of a new-born baby may turn out to be male or female. In these cases, the individual outcomes are uncertain. With experience and enough repetition, however, a regular pattern of outcomes can be seen (by which certain predictions can be made).

10. Formulae Of ProbabilityA dice is thrown 1000 times with frequencies for the outcomes 1,2,3,4,5 and 6 :-Ans. Let Ei denote the event of getting outcome i where i=1,2,3,4,5,6:- Then; Probability of outcome 1= Frequency of 1 Total no. of outcomes = 179 1000 = 0.179 Therefore, the sum of all the probabilities , i.e, E1 + E2 + E3+ E4+ E5 + E6 is equal to 1……….

11. To know the opinion of students for the subject maths, a survey of 200 students was conducted :- Find the probability for both the opinions :Ans. Total no. of observations = 200 P(Likeness of the students) = No. of students who like Total no. of students = 135 200 = 0.675 P(Dislikeness of the students) = 65 200 = 0.325

12. Random Phenomenon An event or phenomenon is called random if individual outcomes are uncertain but there is, however, a regular distribution of relative frequencies in a large number of repetitions. For example, after tossing a coin a significant number of times, it can be seen that about half the time, the coin lands on the head side and about half the time it lands on the tail side.Note of interest: At around 1900, an English statistician named Karl Pearson literally tossed a coin 24,000 times resulting in 12,012 heads thus having a relative frequency of 0.5005 (His results were only 12 tosses off from being perfect!).

13. ApplicationsTwo major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets.

14. A significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, utilize reliability theory in the design of the product in order to reduce the probability of failure. The probability of failure may be closely associated with the product's warranty.Monty Hall ProblemYou're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1The host, who knows what's behind the doors, opens another door, say No. 3, which has a goat.Do you want to pick door No. 2 instead?