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.
Chance• Chance is how likely it is that something willhappen. 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%
Examples Of Chance When 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. 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
The History of ProbabilityProbability originated from the study of games of chance. Tossinga dice or spinning a roulette wheel are examples of deliberaterandomization that are similar to random sampling. Games ofchance were not studied by mathematicians until the sixteenthand seventeenth centuries. Probability theory as a branch ofmathematics arose in the seventeenth century when Frenchgamblers asked Blaise Pascal and Pierre de Fermat (both wellknown pioneers in mathematics) for help in their gambling. In theeighteenth and nineteenth centuries, careful measurements inastronomy and surveying led to further advances in probability.
Modern Use Of ProbabilityIn the twentieth century probability is used to control theflow of traffic through a highway system, a telephoneinterchange, or a computer processor; find the genetic makeupof individuals or populations; figure out the energy states ofsubatomic particles; Estimate the spread of rumors; andpredict the rate of return in risky investments.
Predictable and Unpredictable OccurrencePredictable Occurrences:The time an object takes to hit the ground from a certain heightcan easily be predicted using simple physics. The position ofasteroids in three years from now can also be predicted usingadvanced technology.Unpredictable Occurrences:Not everything in life, however, can be predicted using scienceand technology. For example, a toss of a coin may result ineither a head or a tail. Also, the sex of a new-born baby mayturn out to be male or female. In these cases, the individualoutcomes are uncertain. With experience and enoughrepetition, however, a regular pattern of outcomes can be seen(by which certain predictions can be made).
Formulae Of Probability• A dice is thrown 1000 times with frequencies for the outcomes 1,2,3,4,5 and 6 :- Outcome 1 2 3 4 5 6 Frequency 179 150 157 149 175 190Ans. 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……….
• To know the opinion of students for the subject maths, a survey of 200 students was conducted :- Opinion No. of students Likeness 135 Dislikeness 65 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
Random PhenomenonAn event or phenomenon is called random ifindividual outcomes are uncertain but thereis, however, a regular distribution of relativefrequencies in a large number of repetitions. Forexample, after tossing a coin a significant numberof times, it can be seen that about half thetime, the coin lands on the head side and abouthalf the time it lands on the tail side.Note of interest: At around 1900, an Englishstatistician named Karl Pearson literally tossed acoin 24,000 times resulting in 12,012 heads thushaving a relative frequency of 0.5005 (His resultswere only 12 tosses off from being perfect!).
ApplicationsTwo major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets.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 products warranty.
Monty Hall Problem• Youre given the choice of three doors: Behind one door is a car; behind the others, goats.• You pick a door, say No. 1• The host, who knows whats behind the doors, opens another door, say No. 3, which has a goat.• Do you want to pick door No. 2 instead?
Host reveals Goat A orHost reveals Goat B Host must reveal Goat B Host must reveal Goat A