Unit 8: Probability Theory

Data is handled numerically or in terms of graphs, and information (average size. spread of

data, etc.) is extracted from them.

If these data are influenced by "chance," by factors whose effect we cannot predict exactly

(e.g., weather Jata, stock prices, lifespans of tires, etc.) we rely on probability theory.

Probability theory gives mathematical models of chance processes called random

experiments.

Data Representation, Average and Spread:

(a) Sorting: Suppose there is a data given below: These are n = 14 measurements of the tensile strength of

sheet steel in kg/mm2, recorded in the order obtained and rounded to integer values.

(1) 89 84 87 81 89 86 91 90 78 89 87 99 83 89.

After sorting the values are:

(2) 78 81 83 84 86 87 87 89 89 89 89 90 91 99.

(b) Graphical Representation of data:

Data is presented in following ways:

(1) Steam and Leaf Plot

(2) Histogram

(3) Centre and Spread of Data (Median) and Quartiles

(4) Box plot

(5) Outliers

(1) Steam and Leaf plot:

Suppose there is a data: (1) 89 84 87 81 89 86 91 90 78 89 87 99 83 89.

After sorting we get: (2) 78 81 83 84 86 87 87 89 89 89 89 90 91 99.

Now we group them in ranges as below

Range Frequency Stem Values in Range Leaf 75-79 1 7 78 8

80-84 3 8 81 83 84 134

85-89, 7 8 86 87 87 89 89 89 89 6779999

90-94 2 9 90 91 01

95-99 1 9 99 9

Stem are the integers in Tens position the one which are highlighted in table of Range

Leaf means the last digits of the values in particular range

(2) Histogram:

Better displaying for large set of data

Range Frequency

Class

interval

Class marks

Midpoints

(X-axis)

Relative Frequency

(Y axis)

Frequency/n 75-79 1 74.5

79.5

74.5 + 79.5= 77 1/14 = 0.07

80-84 3 80.5 84.5

82 3/14 = 0.21

85-89, 7 85.5 89.5

87 7/14 = 0.5

90-94 2 90.5 94.5

92 2/14 = 0.14

95-99 1 95.5 99.5

97 1/14 = 0.07

Hence the histogram is as below:

(3) Centre(median) and Spread (Range) of Data and Quartiles

Median: Centre of data values

78 81 83 84 86 87 87 89 89 89 89 90 91 99.

N = 14, the 7th

value is 87 and the 8th

value is 89

Median = (87 + 89)/2 = 88

Range: Max value Min value = 99 78 = 21

Instead of Range, Quartiles give better information.

Inter Quartile Range = Upper Quartile (qu) Lower Quartile (ql)

Upper Quartile: Middle value among data values above median

. 7 plus 8 . 78 81 83 84 86 87 87

means 84 is lower quartile (ql) Lower Quartile: middle value among data values below median

And fourth value from last i.e. 89 = qu i.e. upper quartile

Thus 89 84 = 5 is middle quartile (qm). (4) Box plot

Box plot is obtained from 5 values we have calculated i.e. xmin, ql, qm, qu and xmax.

Box is drawn from ql to qu including qm.

The lines above and below box are xmin to box and xmax to box

For following data:

78 81 83 84 86 87 87 89 89 89 89 90 91 99.

Our xmin =78

Xmax = 99

ql = 84

qu = 89

Box plot of above data is shown below:

(5) Outliers

IQR = 5

1.5 IQR = 5X1.5=7.5

ql - 1.5IQR

84-7.5 = 76.5

And

qu + 1.5IQR

89 + 7.5 = 96.5

See values in the given data 78 81 83 84 86 87 87 89 89 89 89 90 91 99

Only 99 is not in the range of 76.5 to 96.5

So 99 is an outlier

Mean:

Spread can be found out more nicely by standard deviation and Variance.

Probability Theory:

Result of trial is outcome or sample point.

Sample space of an experiment is set of all possible outcomes.

A, B and C are events as shown above

Probability of coming 5 or 6 i.e. 2 numbers so

Probability of coming even number i.e. there are three even numbers from 1 to 6 namely 2, 4

and 6 so

Solution:

There are five coins and each has two outcomes head or a tail so

Binomial Distribution:

Poisson Distribution:

From above table of A7

From above table of A7