In part3 we covered measures of central tendency and measures of variation
In this part we will be looking into:
- Sample spaces and Probability
- Addition Rules for Probability
- Multiplication Rules
- Conditional Probability
- Counting Rules
Probability is the base of inferential statistics. Like, predictions are based on probability and hypothesis are tested by using probability.
1. Sample Spaces and Probability
A sample space is the set of all possible outcomes of a probability experiment.
Example: All possible weather conditions in a day sunny ,rainy ,windy ,stormy.
Probability can be defined as the chance of an event occurring.
Example : Probability of a rainy weather on a particular day is 1 out of 4 possible outcomes where we can say there is 25% chance that it is going to rain.
An event consists of a set of outcomes of a probability experiment.
An outcome is the result of a single trial also called simple event.
Example: The a die is rolled and a 5 shows ,this result is called an outcome.
The event of getting an even number when a die is rolled is called a compound event , since it consists of 3 outcomes or 3 simple events.
There are three basic interpretations of probability:
1. Classical probability
2. Empirical or relative frequency probability
3. Subjective probability
Classical probability assumes that all outcomes in the sample space are equally likely to occur. You don’t have to conduct any experiment to determine the probability. For example, when a die is rolled each outcome has same probability of occurring. Since there are 6 outcomes,each outcome has a probability 1/6.
Problem:
There are four basic probability rules to be followed while solving probability problems:
Rule 1: Probability cannot be negative or greater than 1.
Rule 2 : If an event cannot occur then the probability of that event is 0.
Example : Drawing a black diamond from a deck of cards.
Rule 3: If an event is certain then its probability is 1.
Example : Picking either black or red card from a deck.
Rule 4: The sum of probabilities of all the outcomes in the sample space is always 1.
For example, in the roll of a fair die, each outcome in the sample space has a probability of 1/6. Hence, the sum of the probabilities of the outcomes is 1/6 * 6 times which is 1.
Complementary Events :The complementary of an event E is the set of outcomes in the sample space that are not included in the outcomes of an event E. The complement of E is denoted by “E bar”.
Lets say an event of selecting a day of the week and getting a weekday.
The complement of the above event is getting a Saturday or a Sunday.
Problem:
Empirical Probability relies on actual experience to determine the likelihood of outcomes. In empirical probability, one might actually roll a given die 6000 times,observe the various frequencies, and use these frequencies to determine the probability of an outcome.
Suppose, for example, that a researcher for the American Automobile Association(AAA) asked 50 people who plan to travel over the Thanksgiving holiday how they will get to their destination. The results can be categorized in a frequency distribution as shown.
Now probabilities can be computed for various categories. For example, the probability of selecting a person who is driving is 41/50 , since 41 out of the 50 people said that they were driving.
Empirical Probability P(E) = frequency for the class/total frequencies in the distribution
Law of Large Numbers
If the empirical probability of getting a head is computed by using a small number of trials, it is usually not exactly . However, as the number of trials increases, the empirical probability of getting a head will approach the theoretical probability of , if in fact the coin is fair (i.e., balanced). This phenomenon is an example of the law of large numbers.
Subjective Probability uses a probability value based on an educated guess or estimate, employing opinions and inexact information,a doctors view on a patients health condition.
2. The Addition Rules for Probability
Addition Rule 1
When two events A and B are mutually exclusive, the probability that A or B will occur is:
P(A or B) = P(A) + P(B)
Addition Rule 2
If A and B are not mutually exclusive, then
P(A or B) = P(A) + P(B) -P(A and B)
3. The Multiplication Rules
Two events A and B are independent events if the fact that A occurs does not affect the probability of B occurring.
Multiplication Rule 1
When two events are independent, the probability of both occurring is:
P(A and B) = P(A) . P(B)
4. Conditional Probability
When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed, the events are said to be dependent events.
For the problem above discussed, the probability of getting an ace on the first draw is 4/52 , and the probability of getting a king on the second draw is 4/51 in case of dependent event . By the multiplication rule, the probability of both events occurring is 4/52 * 4/51
The event of getting a king on the second draw given that an ace was drawn the first time is called a conditional probability.
The conditional probability of an event B in relationship to an event A is the probability that event B occurs after event A has already occurred.
Multiplication Rule 2
When two events are dependent, the probability of both occurring is:
P(A and B) = P(A) . P(B|A)
5. Counting Rules
Fundamental Counting Rule
In a sequence of “n” events in which the first one has k1 possibilities and second has k2 possibilities and so on, the total number of possibilities for the sequence will be k1. k2 . k3 … kn.
Example:
A coin is tossed and a die is rolled. Find the number of outcomes for the sequence of events.
Solution:
The coin can land either heads up or tails up and the die can land with any
one of six numbers showing face up. So there are 2 * 6 12 possibilities.
Permutations
A permutation is an arrangement of n objects in a specific order.
Combinations
A selection of distinct objects without regard to order is called a combination.
