The class on 18th July 2013 kicked off with
the introduction to probability. Three methods of probability are:
- Ratio Chance
- Percentage Chance
- Probability
Percentage Chance
Another way to talk about probability is by
using percents. Percent means "out of 100." Instead of describing a
probability as 1 out of 4, we say 25%. Think of it this way: 25 out of 100 is
one quarter of 100, just as 1 is one quarter of 4.
- Turning left and turning right are Mutually Exclusive (you can't do both at the same time)
- Tossing a coin: Heads and Tails are Mutually Exclusive
- Cards: Kings and Aces are Mutually Exclusive
1.
"1 out of 4" = 1/4
2.
Divide the top number by the bottom number 1 ÷ 4
= 0.25
3.
Then, move the decimal 2 places to the right and
add the percent sign
0.25 = 25%
Probability
Probability is the
chance that a specific event will happen. Measuring probability helps us
understand everything from health risks to weather reports to baseball.
For example:
The referee at a
basketball game tosses a coin to decide which team starts. Your team picks
“heads.” You've got a 1 out of 2 chance of winning the toss, since the coin can
land 1 of 2 ways (either heads or tails).
3 Approaches
The three main approaches
used for probability are as follows:
1.
A Priori
(prior information)
Probability calculated by
logically examining existing information. A priori probability can most easily
be described as making a conclusion based upon deductive reasoning rather than
research or calculation. The largest drawback to this method of defining
probabilities is that it can only be applied to a finite set of events.
For example:
Consider how the price of a share
can change. Its price can increase, decrease or remain the same. Therefore,
according to a priori probability, we can assume that there is a 1-in-3, or
33%, chance of one of the outcomes occurring (all else remaining equal).
2.
Empirical (Information Collected)
Empirical approach is a method
that determines probability from data on actual experiments in order to
determine approximate probabilities. Under this method probability is
defined as the frequency of occurrence of an event N (A), to the number of
trials in the experiment, N. This is represented as P (A) = N (A)/N. It may be
noted that the probability, as defined above, is only a ratio of two numbers,
in which the numerator N (A) is the number of favorable cases and N is the
number of possible outcome satisfying certain conditions.
In other words, imagine tossing
the die 100 times, 1000 times, 10,000 times, ... . Each time we expect to get a
better and better approximation to the true probability of the event A. The
mathematical way of describing this is that the true probability is the
limit of the approximations, as the number of tosses "approaches
infinity
3.
Subjective (Judgement)
A probability
derived from an individual's personal judgment about whether a specific outcome
is likely to occur. Subjective probabilities contain no formal calculations and
only reflect the subject's opinions and past experience.
The sample space is
an exhaustive list of all the possible outcomes of an experiment. Each possible result of such a study is represented by one and only one point in the sample
space, which is usuall denoted by S.
Examples:
Experiment Tossing a
coin:
Sample space S = {Heads,Tails}
Experiment Measuring
the height (cms) of a girl on her first day at school:
Sample space S = the set of all possible real numbers
Venn diagrams
An illustration that uses
overlapping or non-overlapping circles to show the relationship between finite
groups of things. The circles overlap, items have a
specified something in common. Where the circles do not overlap, the items do
not have a specified something in common
Venn Diagram with two variables
Venn Diagram with three variables
Venn Diagrams with four variables
3 Event Properties
1.
Mutually Exclusive Events
2.
Independent Events
3.
Dependent Events
Mutually Exclusive Events
Events that can’t
happen at the same time.
Example:
- Turning left and turning right are Mutually Exclusive (you can't do both at the same time)
- Tossing a coin: Heads and Tails are Mutually Exclusive
- Cards: Kings and Aces are Mutually Exclusive
Independent Events
Events that are not affected by
the occurrence of the previous events are called independent events.
For Example:
Two separate tosses of a fair
coin are independent events. The result of the first toss has no effect on the
probability of heads or tails on the second toss.
Dependent Events
If the result of one event is affected by the result of another event, the events are said to be dependent events
For Example:
A drawer
contains 3 red paperclips, 4 green paperclips, and 5 blue paperclips. One
paperclip is taken from the drawer and is not replaced.
Another paperclip is taken from the drawer. Because the first paper clip is not replaced, the sample space of the
second event is changed. The sample space of the first event is 12
paperclips, but the sample space of the second event is now 11
paperclips. The events are dependent.
3 Laws of
Probability
Addition Law
When two events,
A and B, are mutually exclusive, the probability that A or B will occur is the
sum of the probability of each event.
P(A or B) = P(A)
+ P(B)
Conditional Law
Probability of
an event or outcome based on the occurrence of a previous event or outcome.
Conditional probability is calculated by multiplying the probability of the
preceding event by the updated probability of the succeeding event
The conditional
probability of B occurring given that event A has occurred is written P(B/A).
P(B/A) = P(A∩B)/P(A)
Multiplication
Law
This law for probabilities states that if A and B are
independent events then
P(A ∩ B)=P(A)×P(B),
and, in the case of n independent events, A1, A2,…, An,
P(A1 ∩ A2 ∩…∩ An)=P(A1)×P(A2)×…×P(An).
The class concluded with this topic.
Neeraj Garg
Pallavi Gupta
Pallavi Gupta
Piyush
Prerna Bansal
Priya Jain
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