Unit -III: Concept of probability

Unit - III: Concept of probability: Probability distribution - normal, poisson, binomial; descriptive

statistics - central tendency, dispersion, skewness and kurtosis.


Sampling: Probability and non-probability

Q 1: What is Probability? Why is it used in psychological research?

Probability:

Suppose there are 25 balls in an urn. 8 among them are blue, 9 are red and 7 are green in color. If now three balls are taken out randomly, what is the likelihood of occurrence that each of the three balls will be of three different colors? The measure of this likelihood of occurrence of an event is called probability. The higher the degree of probability, the more likely the event is to happen, or, in a longer series of samples, the greater the number of times such event is expected to happen. There are a few underlying assumption of probability. These are:

• Equally likely: Elementary events have the same chance of occurrence.
• Mutually Exclusive: If any one of the elementary events occurs then none of the remaining elementary units can occur.
• Elementary events are collectively exhaustive: When taken together they are collectively exhaustive in the sense that no other outcome can occur. Probability is then 1.

Number of ways it can happen

Total number of outcomes

Therefore,

Probability of an event happening =

Example:

There are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble will be picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability =	4/5  = 0.8

Probability in Psychological research:

The concept of probability is relevant to experiments that have somehow uncertain outcomes. There can be situations where despite every effort to maintain fixed (identical conditions) some variations of the result in repeated trials of the experiment are unavoidable. Psychology as a domain of social science deals with individual human being and their behavior, none of which are definite or certain. Consequently psychological research largely involves varied results and findings as because behavior can never be absolutely certain. For example, suppose a child has an IQ of 150. But can we say that the child will definitely score full marks in mathematics? An individual has low trait anxiety. But does it mean that the individual will not be anxious in case of an emergency? So, one of the chief drawbacks of any branch of social scientific research is the presence of several other extraneous variables in addition to the independent and dependent variables. This renders the uncertain nature of psychology. And this is where the concept of probability creeps in. Thus to measure or assess any psychological construct we need the concept of probability in order to make sense out of the distribution of the construct in the larger population. Thus probability is very much required in psychological research.

Q 2: What is Probability Distribution? Describe the characteristics of normal probability distribution.

A probability distribution is a distribution of probabilities of occurrence of scores, events or cases among the classes of the given variable. Probabilities of occurrence of scores can be obtained when the n is large. In such a case the relative frequency (f/n) of scores can be an estimate of their probability of occurrence. In this way a probability distribution may be computed experimentally, using the observed frequencies of scores or events in the data of a test or experiment.

       Theoretically probability distributions, on the contrary, are computed theoretically on the basis of specific mathematical models and laws of probability. They are used widely in predicting probabilities of events and in testing hypothesis experimental hypotheses. Example includes the normal distribution computed on the basis of Gaussian equation. In any probability distribution probabilities are in continuous scale with no real gaps between them. But the scale for events, cases or scores of the variable under consideration may be either continuous or discontinuous. Thus a probability distribution of scores of a discontinuous variable has real gaps in its scale for the scores or cases, and is called a discrete probability distribution. Examples of such distributions are binomial distribution and Poisson distribution. On the other hand a distribution with no real gap in its scale of scores is called a continuous probability distribution such as normal distribution and t- distribution.

Characteristics:

1. The normal probability curve is a bell shaped curve with a single peak at the exact center of the distribution. Thus it is a unimodal distribution.
2. The normal curve is symmetrical about its mean such that the area on the left side of the curve is equal to the right side of the curve.
3. For a normal curve, the arithmetic mean, median and mode lie at the same point i.e. at the centre.
4. The normal probability distribution is asymptotic i.e. the both tails of the curve gets closer and closer to x-axis but touches it.
5. The normal probability distribution is a standardized one i.e. it has a mean of 0 and a standard deviation of 1. Thus for two different standardized normal distributions, any particular observation can be compared by converting the raw score into z-value. Z-value can be obtained by the following formula,

6. The total area under the normal curve is 1. Between the mean and ±3σ of the normal curve, 99.7 % of the total area falls while that between mean and ±2σ is 95.4% and for ±1σ is 68.26%.      

7.   The normal curve follows the Gaussian Equation. The normal distribution is

8.  The normal curve has two inflection points (where the second derivative of f is zero and changes sign), located one standard deviation away from the mean, namely at x = μ − σ and x = μ + σ.

9.   The normal curve has zero skewness as also zero kurtosis. Skewness is the measure of the extent to which a probability distribution of a real-valued random variable "leans" to one side of the mean. It can be positive or negative. On the other hand, kurtosis is any measure of the "peakedness" of the probability distribution of a real-valued random variable. It can be platykurtic (flat) or leptokurtic (too peaked). A normal curve is mesokurtic in nature (no peakedness).

The Poisson Probability Distribution

The Poisson Distribution was developed by the French mathematician Simeon Denis Poisson in 1837.

The Poisson random variable satisfies the following conditions:

1. The number of successes in two disjoint time intervals is independent.
2. The probability of a success during a small time interval is proportional to the entire length of the time interval.

Apart from disjoint time intervals, the Poisson random variable also applies to disjoint regions of space.

Applications

• the number of deaths by horse kicking in the Prussian army (first application)
• birth defects and genetic mutations
• rare diseases (like Leukemia, but not AIDS because it is infectious and so not independent) - especially in legal cases
• car accidents
• traffic flow and ideal gap distance
• number of typing errors on a page
• hairs found in McDonald's hamburgers
• spread of an endangered animal in Africa
• failure of a machine in one month

The probability distribution of a Poisson random variable X representing the number of successes occurring in a given time interval or a specified region of space is given by the formula:

P(X)=​x!​​e​−μ​​μ​x​​​​

where

x=0,1,2,3…
e=2.71828 (but use your calculator's e button)
μ= mean number of successes in the given time interval or region of space

Mean and Variance of Poisson Distribution

If μ is the average number of successes occurring in a given time interval or region in the Poisson distribution, then the mean and the variance of the Poisson distribution are both equal to μ.

E(X) = μ

and

V(X) = σ² = μ

Note: In a Poisson distribution, only one parameter, μ is needed to determine the probability of an event.

Example 1

A life insurance salesman sells on the average 3 life insurance policies per week. Use Poisson's law to calculate the probability that in a given week he will sell

1. Some policies
2. 2 or more policies but less than 5 policies.
3. Assuming that there are 5 working days per week, what is the probability that in a given day he will sell one policy?

Here, μ = 3

(a) "Some policies" means "1 or more policies". We can work this out by finding 1 minus the "zero policies" probability:

P(X > 0) = 1 − P(x₀)

Now P(X)=​x!​​e​−μ​​μ​x​​​​ so P(x​0​​)=​0!​​e​−3​​3​0​​​​=4.9787×10​−2​​

Therefore the probability of 1 or more policies is given by:

Probability=P(X≥0)

=1−P(x​0​​)

=1−4.9787×10​−10​​

=0.95021

(b) The probability of selling 2 or more, but less than 5 policies is:

P(2≤X<5)

=P(x​2​​)+P(x​3​​)+P(x​4​​) 

=​2!​​e​−3​​3​2​​​​+​3!​​e​−3​​3​3​​​​+​4!​​e​−3​​3​4​​​​
=0.61611

(c) Average number of policies sold per day: ​5​​3​​=0.6

So on a given day, P(X)=​1!​​e​−0.6​​(0.6)​1​​​​=0.32929

For reference: http://www.intmath.com/counting-probability/13-poisson-probability-distribution.php