A topic of Biostatistics

1. Probability
Prof. (Dr) Shahla Yasmin
Dept of Zoology
Patna Women’s College

2. Probability and its rules
• Probability is the quantification of chance; is Usually expressed as
symbol ‘p’
• Probability ‘p’ ranges from 0 to 1, sometimes expressed as %ages
• P=0 means ‘ no chance of an event happening’
• P=1 means ‘100% chances of an event happening’
• If probability of an event happening is ‘p’ and probability of not
happening is ‘q’ , then, p + q = 1
• p = no. of events occurring/ total no. of trials
• If an experiment is repeated ‘n’ times and an event A is observed
‘f’ times p(A) = f/n
• There are two main rules associated with basic probability: the
addition rule, and the multiplication rule,

3. Addition rule of probability
• If one event excludes the possibility of occurrence of the other
specified event or events, the events are called mutually exclusive.
• e.g. Getting head excludes the possibility of getting tail in coin flip
• e.g. Birth of a male child excludes the possibility of birth a female
• These mutually exclusive events follow the addition law of
probability.
• P(A or B) = P(A) + P(B)
• e.g. in a dice P(2) = 1/6
• P(5) = 1/6
• P(2 or 5) = 1/6 + 1/6 = 1/3

4. Multiplication law of probability
• It is applied when two or more events are occurring together
but they are independent of each other.
• P(A and B) = P(A) * P(B)
• e.g in a dice
• P(2) = 1/6
• P(5) = 1/6
• P(2 and 5) = 1/6 * 1/6 = 1/36

5. Multiplication law of probability
• If we assume twin pregnancy will occur once in 80 pregnancy.
And child with Rh-ve blood group will be born once in 10
births.
• probability of male child, p1 = ½
• probability of child being Rh+ve, p2 = 9/10
• probability of single birth, p3 = 79/80
• probability of male, Rh+ve and single birth
= ½*9/10*79/80 = 711/1600

6. Binomial Probability distribution
• A probability distribution, in which the number of possible
outcomes from a single trial is two (e.g. male or female) is
called binomial distribution.
• Binomial probability distribution is formed by terms of the
expansion of binomial expression (p +q)n
• n= number of events
• p= probability of success
• q= probability of failure
• when n = 2 ,( p + q )2 = p2 + q2 + 2pq
• when n = 3 ( p + q )3 = p3 + q3 + 3pq2 + 3p2q
• when n = 4 ( p + q )4 = p4 + 4p3q+ 6p2q2 + 4pq3 + q4

7. Binomial probability distribution
• 20% of children under 6 yrs of age were found to be severely
malnourished. Only 4 children were selected at random. What is
the probability of 4, 3, 2, 1, 0 being severely malnourished?
• p = 0.2, q = 0.8, n = 4
• ( p + q )4 = p4 + 4p3q+ 6p2q2 + 4pq3 + q4
• = 0.24 + 4*0.23*0.8+ 6*0.22*0.82 + 4*0.2*0.83 + 0.84
• = 0.0016 + 0.0256 + 0.1536 + 0.4096 + 0.4096
• Probability of all 4 children being malnourished = 0.16%
• Probability of 3 children being malnourished = 2.56%
• Probability of 2 children being malnourished = 15.36%
• Probability of 1 children being malnourished = 40.96%
• Probability of no children being malnourished = 40.96%

8. Binomial Probability distribution histogram is perfectly symmetrical

9. Poisson probability distribution
• Poisson distribution is a limiting form of the binomial
distribution in which n, the number of trials,
becomes very large & p, the probability of success of
the event is very very small.
• The distribution was derived by the French
mathematician Siméon Poisson in 1837, and the first
application was the description of the number of
deaths by horse kicking in the Prussian army.

10. Poisson probability distribution
• Poisson distribution used in cases where the chance of any
individual event being a success is very small. The distribution
is used to describe the behaviour of rare events.
• Examples;
• The number of defective screws per box of 5000 screws.
• The number of printing mistakes in each page of the first
proof of book.
• The number of air accidents in India in one year.

11. • P(X) = e-μ μx/X!
• Where, X = 1,2,3,4….. e = 2.7183 (the base of
natural logarithms)
• μ = the mean of Poisson distribution i.e. the
average number of occurrence of an event.

12. Condition Under Which Poisson
Distribution is Used
• The random variable X should be discrete.
• A dichotomy exists, i.e. happening of the
event must be of two alternatives such as
success & failure.
• Applicable in those cases where the number
of trials n is very large and the probability of
success p is very small

13. Problem
• The average number of accidents at a
particular intersection every year is 18.
• (a) Calculate the probability that there are
exactly 2 accidents there this month.
• There are 12 months in a year, so 12 /18 = 1.5
accidents per month
• P(X) = e-μ μx/X!
• P(X = 2)= e-1.5 1.52/2! = 0.2510

14. Poisson probability distribution histogram is skewed

15. Normal probability distribution
• Normal (or Gaussian ) distribution is defined
as a continuous frequency distribution of
infinite range.
• The normal distribution is a descriptive model
that describes real world situations
• Characteristic Bell-Shaped curve is obtained
• Mean=Median=Mode

16. Normal probability histogram are gradual because of large
sample size and measurement on a continuous scale

17. Normal probability distribution
• In binomial probability distribution, the
number of possible outcomes from a single
trial is two. The probability of each alternative
outcome is equal (i.e. 0.5) and the probability
distribution is perfectly symmetrical.
• When the number of trials become large, the
shape of probability distribution changes
towards normal probability distribution.

18. Normal probability distribution
• If a large number of measurements is made at an infinitely high
degree of precision the steps in the histogram are transformed
into a smooth, continuous curve .
• A Normal curve is symmetrical, with the axis of symmetry
passing through the mean on the baseline
• There is point of inflection on each side of the curve, where the
curve changes from convex to concave
• The distance of point of inflection from the central axis can be
used as a standard unit of distance by which the baseline is
divided into equal segments. This one standard unit of distance
is usually called one standard deviation

19. Normal probability distribution
• The area bounded by one standard deviation on either side of
the axis of symmetry (mean) is approx 68.26% of the total
area
• This means in large samples of Normally distributed data
about 68% of the observations fall within one SD on either
side of the mean
• The remaining 32% of the observations fall outside these
limits ( 16% above 1SD and 16% below 1SD). These are called
z scores. When z= ±2, proportion of observations included by
the curve = 95.44%. When z= ±3, proportion of observations
included in the curve = 99.74

20. Point of inflection
Divisions of normal distribution curve

21. • Most characters such as height or mass and
even eye colour, show continuous variation
as shown in the graph on the following slide

22. 7
6
5
4
3
2
1
0
Negro Dark Mulleto Fair Albino
Graphical representation of polygenic inheritance of skin colour in
human 22

23. Properties of Normal Distributions
1. The mean, median, and mode are equal.
2. The normal curve is bell-shaped and is symmetric about the
mean.
3. The total area under the normal curve is equal to 1.
4. The normal curve approaches, but never touches, the x-axis
as it extends farther and farther away from the mean.
x
Total area = 1
μ

24. • A normal distribution can have any mean
and any positive standard deviation.
• The mean gives the location of the line of
symmetry.
• The standard deviation describes the
spread of the data.
Between μ – σ and μ+σ (in the
center of the curve), the
graph curves downward.
The graph curves upward
to the left of μ – σ and to
the right of μ + σ. The
points at which the curve
changes from curving
upward to curving
downward are called the
inflection points.

25. Example: Understanding Mean and
Standard Deviation
1. Which normal curve has the greater mean?
Solution:
Curve A has the greater mean (The line of symmetry
of curve A occurs at x = 15. The line of symmetry of
curve B occurs at x = 12.)

26. Example: Understanding Mean and
Standard Deviation
2. Which curve has the greater standard deviation?
Solution:
Curve B has the greater standard deviation (Curve
B is more spread out than curve A.)

28. • The Empirical Rule:
About 68% of the area
under the graph is within
one standard deviation of
the mean;
• about 95% of the area
under the graph is within
two standard deviations
of the mean;
• about 99.7% of the area
under the graph is within
three standard deviations
of the mean.