Probability in Biostat

1. PROBABILITY
Ms. Chaitali C. Dongaonkar

2. ▪ Measure of likelihood that an event will occur in random experiment
▪ Examples: Prediction about New drug, Estimation of production
costs, forecasting vaccine failure
▪ Element of chance
▪ Ratio of number of favourable outcomes to the all possible outcomes
i.e. favourable outcomes + unfavourable outcomes
▪ Let, “a” is number of favourable outcomes of an event “A” and “b” is
number of unfavourable outcomes of an event “A”
What will be the Probability of occurrence of A….???

3. ▪𝑃 𝐴 = Number of favourable outcomes/Total Number of outcomes
▪𝑃 𝐴 = a/a+b
▪ The Probability of unfavourable outcomes of A is given by,
▪ = Number of unfavourable outcomes/Total Number of outcomes
▪ = b/a+b
▪ Total Probability = a/a+b + b/a+b = 1
▪ Theoretically, Total probability = 1
▪ Practically, 0≤ P ≤ 1
▪ Probability of event: ratio, fraction, percentage.

4. ▪ Experiment or Trial: any procedure that can be infinitely repeated
▪ List of outcomes specified in advance
▪ Actual occurrence of outcome cannot be predicted in advance.
▪ Example: Probability of getting 5 on top of die.
▪ Sample Space: List of all possible outcomes of an experiments
▪ Example: if we throw die then sample space consists of { 1,
2, 3,4,5,6 }
▪ Event: The occurrence of particular outcome or combinations of outcomes
▪ Example: Tossing of coin, outcome of this will be “Head” or “Tail”

5. 1. In plastic container, there are 5 Crocin and 10 Aspirin Tablets. If
one tablet is chosen at random, find the probability that Aspirin
tablets or Crocin tablets are selected.

6. ▪ Measure of probability of an event occurring
▪ Another event has already occurred

7. Probability Distribution
Discrete Probability
Distribution
1. Binomial Distribution
2. Poisson Distribution
Continuous Probability
Distribution
1. Uniform Distribution
2. Exponential Distribution
3. Normal Distribution

9. ▪ Also known as Bernoulli's distribution
▪ Simply success or failure outcomes when experiments repeated several
times
▪ Two possible outcomes, Prefix “bi” = Two/Twice
▪ Examples: Coin tossed two possible outcomes Head or Tail, Taking test
two possible outcomes Pass or Fail, Lottery ticket Win money or Not
▪ Only success or failure represented by “Binomial Distribution”
▪ Success (p) and Failure (q), p + q = 1 (0.5 + 0.5 = 1)
▪ P (x success in n trials)= nCXPXqn-x
▪ n is number of trials, p is probability of success in trial, q is probability of
failure in trial

10. ▪ Fixed number of observations or trials
▪ Each trial results as outcome as Success (s) or Failure (f)
▪ Each observation or trial is independent
▪ Probability of success exactly same from one trial to another i.e. DOES
NOT CHANGE WITH TIME
▪ Mean 𝑥
¯
= np, Variance 𝜎2 = npq, Standard Deviation 𝜎 = 𝑛𝑝𝑞
▪ where, n is number of trials, p is probability of success in trial, q is
probability of failure in trial
▪ Mean shows average number of success, variance less than its mean
▪ p = q, symmetrical distribution, p < 0.5 = + ve skewness, p>0.5 = - ve
skewness

11. 1. The mean of binomial distribution is 40 and its standard deviation
is 6. Calculate n, p, q.

12. 2. Multipunch tablet machine produces 12% defective tablets. What
is the probability that out of random sample of 20 tablets produced
by the machine, 4 are defective ?

14. 𝑛𝐶𝑟
=
𝑛!
n − r ! ∗ r!

15. ▪ Simeon Denis Poisson – in 1837, French mathematician
▪ For Number of discrete events in given period of time (fixed)
▪ Trials = large i.e. n is very large
▪ Success = very small i.e. 0
▪ Used for rare events like number of people arriving in 1 hr, number
of phone calls in a day, defective dosage per batch, number of
microbes in samples
▪ Represents number of occurrence of an event in one unit of time

17. ▪ Events are discrete i.e. you can count them
▪ Events can not happen at the same time
▪ Events are independent
▪ Probability of two or more occurrences in a very small interval is
close to Zero
▪ More symmetric as its mean or variance increases
▪ Uniparametric in nature
▪ Approximation for binomial distribution

18. 1. If the probability that an individual suffers a adverse reaction
from a swine flu vaccine injection is 0.002, determine the
probability that out of 1000 individuals (i) exactly two, (ii) at most
two will suffer a adverse reaction.

21. ▪ Known as normal probability curve or Gaussian distribution
▪ Frequencies distributed evenly about mean of distribution
▪ Univariate distribution used for continuous random variables
▪ Real life situations like monthly salary of employees in a locality,
marks of students in an entrance test
▪ Shape of normal distribution is bell shaped curve

22. ▪ Mean = Mode = Median
▪ Curve is symmetric about the mean i.e. at the centre
▪ Exactly half values are to the left and half values are to the right
▪ The total area under curve is 1 or 100%
▪ Standard normal model = normal distribution with mean of 1 and
standard deviation of 1
▪ Curve: only one top point so its unimodal
▪ No skewness
▪ SD = determines width of the curve
▪ Fitting of actual observed frequency

25. 1. In male student population of 500, the mean height is 70.2
inches and variance is 10.5 inches. How many male students
have height more than 73 inches?
(Given: Area under normal curve corresponding to z = 0.8641 is
0.3039)