The document discusses probability and chance. It defines probability as a measure of how likely an event is to occur from 0 to 1, with 1 being certain and 0 being impossible. Chance is expressed as a percentage, with 50% meaning equally likely. Examples are given of probability in weather forecasting and games. The origins and modern uses of probability are outlined in fields like traffic control, genetics, and investment returns. Predictable versus unpredictable events are distinguished. Formulae for calculating probability from sample data are provided. Random phenomena are described as having uncertain individual outcomes but regular relative frequencies over many repetitions, like coin tosses. Applications in risk assessment and commodity markets are mentioned. Reliability engineering in product design is discussed as using probability of

1. Mathematics
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2. Probability and Chance

3.  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.  Chance 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 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

6. 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.

7. MODERN USE OF PROBABILITY
In 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.

8. Predictable 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).

9. Formulae Of Probability
• A 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……….
Outcome 1 2 3 4 5 6
Frequency 179 150 157 149 175 190

10. • 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
Opinion No. of students
Likeness 135
Dislikeness 65

11. 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!).

12.  Two 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 product's
warranty.

13. Monty Hall Problem
 You're 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 what's behind the doors,
opens another door, say No. 3, which has a goat.
 Do you want to pick door No. 2 instead?

14. Host must
reveal Goat B
Host must
reveal Goat A
Host reveals
Goat A
or
Host reveals
Goat B

16. Madeby:-
Shivansh Jagga
Simar Kohli
Arpit Dash