Lecture Week 3: Probability, Probability

Distribution & Sampling Distribution

Prepared by: Dr. Nurul Syaza Abdul Latif

Probability!

Probability is a likelihood or chance that an event will occur. Example: The chance of picking a red ball from a box

containing red and blue balls, or the chance of raining today !

Definition: If S is the sample space (a set containing all the possibility outcomes of an experiment) and A is an event (a subset of S) then the probability of A is !!!

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Example 1Consider of an experiment in tossing a die. There are six possible outcomes when a die is tossed once. Sample space S={1,2,3,4,5,6} Suppose A is the event of getting odd number i.e. A={1,3,5}. Therefore n(A)=3 !

!

!

Hence, the probability of getting of getting an odd number is 0.5

Venn Diagram is occasionally helpful in determining the probability of simple events

Concepts: Combinations & Permutations

Definition (Combinations): If there are N elements in a population and we want a sample of size r, then the number of ways of selecting r elements from a total of N is given by

In your plant disease management class, you are assigned to read any 4 books from a list of 10 books. How many different groups of 4 are available from the list of 10?

Definition (Permutations): If there are N elements and we want to select r elements but the order in which the elements are selected is IMPORTANT, then the number of ways this can be done is

Example 3: The number of possible ordered seating arrangements for eight people in five chairs.

Example 4: In a lucky draw box contains brand new iPhone 5c. There are 5 yellow, 4 red and 3 blue in that box. In how many ways can Ali choose 4 iPhones from the box? What is the probability Ali chooses 2 yellow iPhones, one red iPhone and one blue iPhone?

Solution Example 4

Probability RulesP(entire sample space)=1 For any events A: 0< P(A) <1 if A’ is the compliment of A, then P(A’)=1-P(A) Events A & B are independent if P(A)=P(A|B) Multiplication rules !!!!!!Conditional probability: !Events A & B mutually exclusive if Addition rules !!!

Example 5: Suppose 20 students share the same floor of a dormitory. Eleven of them took Statistics class, eight took Chemistry class and three took both Statistics and Chemistry classes. A student is chosen at random from the 20 students. What is the probability he took either the Statistics or Chemistry classes? What is the probability he did not take any of two classes?

Solution Example 5

Let theta={3,4,5,6,7,8,9,10,11,12,13,14,15}. If A is the event `an odd number' and B is the event `a multiple of five’, find

Solution Example 6

Example 7: If a coin is toss, what is the probability of getting a tail and a head?

Example 8:

Solution Example 8

Cont… Example 8

Cont… Example 8

2. Probability Distributions

The experiment involves n identical trials or observations. Each trial or observation has only two possible, mutually exclusive outcomes denoted as success or as failure. The outcome of interest to the researcher is labelled a success. Each trial is independent of the previous trial The probability of getting success is p in any one trial, and the probability of getting failure is q=1-p in any one trial. The terms p and q remain constant throughout the experiment.

2.1 Binomial Distributions

Example 9: Given that r ~ b(10,0.2). Find: P(r=0)

!

P(r=2) !

P(r<=2) !

the mean the standard deviation

Example 10: Privacy is a concern for many users of the internet. One survey showed that 59% of internet users are somewhat concerned about the confidentially of their email. Based on this information, what is the probability that a random sample of 10 internet users, 6 are concerned about the privacy of their email?

Solution Example 10

Example 11: A biologist is studying a new hybrid of tomato. It is known that the seeds of the hybrid tomato have probability of 0.70 germinating. The biologist plants 6 seeds. a)What is the probability that EXACTLY four seeds will germinate? b)What is the probability that AT LEAST four will germinate? c)What is the mean and standard deviation?

Solution Example 11

a) EXACTLY four seeds will germinate?

Solution Example 11b) AT LEAST four will germinate?

c) What is the mean and standard deviation?

2.2 Normal Distributions

The normal probability distribution is a continuous probability distribution whose density function has a bell-shaped graph !

The distribution is both symmetrical and mesokurtic. Symmetrical means that each half of the distribution is a mirror image of the other half. Mesokurtic means that it is neither flat nor peaked in terms of the distribution of observed values.

The normal probability distribution is important in statistical inference for the following reasons:

The measurements obtained in many random processes are known to follow this distribution The normal probability distribution fits many human characteristics such as height, speed, IQ, scholastic achievement and years of life expectancy. Living things in nature such as trees, animals, and insects also have many characteristics that are normally distributed. The area under the curve in figure yields the probabilities, so the total area under the curve is 1. !!!!A probability value for a continuous random variable can be determined only for an interval of values. That is

Standard Normal Distribution Because the function is so complex, using it to determine areas under the curve is difficult and time-consuming. Thus table of normal probabilities can be used to analyse normal distribution problems. This is defined by standard normal distribution denoted by z. Standard normal distribution is the normal probability distribution with

Using table to find z

Between 48kg and 55kg

Less than 62kg

3. Sampling DistributionA population consists of the totality of the observations with which we are concerned. For example: the heights of UMK students, the length of crop trees in a plantation area etc. A sample is a subset of a population. The main purpose in selecting random samples is to get information about the unknown population parameters such as the population mean and the population variance. e.g. : Suppose we want to measure the heights of every students in UMK. It would be quite impossible to measure the height of every students in UMK in order to compute the value , representing the population mean. Instead of a large random sample (e.g. specific to only campus jeli) is selected and the mean height for the sample is obtained. The value is now used to make an inference concerning the true mean height . Since many random samples are possible from the same population, we would expect to vary from sample to sample. That is is a value of a random variable that we represent by x . Such random variable is called statistics. Since a statistics is a random variable that depends only on the observed sample, it must have a probability distribution. The probability distribution of a statistic is called a sampling distribution.

a) probability that a single trout taken at random from the pond is between 8 and 12 inches long? !

a) probability that the mean length x ̄of five trout taken at random is between 8 and 12 inches ? !