Basic Probability : Types of Probability, Rules of Addition, Rules of Multiplication, Permutations & Combinations

Probability Concept Identifier

Classical Probability: 

The classical probability formula is used to calculate the probability of an event in a sample space where all outcomes are equally likely. The formula is:

P(E) = Number of favorable outcomes / Total number of possible outcomes

Where:

• P(E) is the probability of the event E

• Number of favorable outcomes is the number of outcomes that correspond to the event E

• Total number of possible outcomes is the number of outcomes in the sample space.

For example, if you roll a fair six-sided dice, the sample space is {1,2,3,4,5,6}, and the probability of rolling a 3 is:

P(rolling a 3) = 1/6 , because there is only one favorable outcome (rolling a 3) out of six possible outcomes.

What are mutually exclusive events?

Mutually exclusive events are events that cannot happen at the same time. If one event occurs, the other event cannot occur simultaneously.

A simple example of mutually exclusive events is tossing a coin, where the events of getting a heads and getting a tails are mutually exclusive, as it's impossible for both events to happen simultaneously.

What are collectively exhaustive events?

Collectively exhaustive events refer to a group of events that includes all possible outcomes of an experiment. This means that at least one of the events must occur every time the experiment is conducted. In other words, the set of events cover all the possibilities, leaving no room for any other outcome.

Collectively Exhaustive Events Example:

An example of collectively exhaustive events is when rolling a standard six-sided die, the set of events {rolling a 1, rolling a 2, rolling a 3, rolling a 4, rolling a 5, rolling a 6} is collectively exhaustive because one of these events must occur every time the die is rolled, and there are no other possible outcomes.

Empirical Probability:

Empirical probability is the chance of something happening based on what has happened before. It's found by dividing the number of times the outcome happened by the total number of times the experiment was done.

What is the Law of Large Numbers?

The Law of Large Numbers says that the more times we repeat an experiment, the closer our results get to what we expect to happen.

The Law of Large Numbers Example:

Another example of the Law of Large Numbers is rolling a fair six-sided die. If you roll the die 10 times, you might get an unusual result such as rolling a 6 four times. But if you roll the die 1,000 times, the proportion of times each number is rolled will be much closer to the expected probability of 1/6 for each number. The more times the die is rolled, the closer the experimental results will be to the theoretical probability of 1/6 for each number.

The Special Rule of Addition

The Special Rule of Addition in probability states that if two events are mutually exclusive, meaning that they cannot both occur at the same time, the probability of either event occurring is the sum of their individual probabilities.

In other words, if we have two events A and B, and they are mutually exclusive, then the probability of either event A or event B occurring is:

P(A or B) = P(A) + P(B)

The Special Rule of Addition Examples:

1. A coin is tossed, and we want to find the probability of getting heads or tails. Since the two outcomes are mutually exclusive, the probability of getting heads or tails is the sum of their individual probabilities, which is 1/2 + 1/2 = 1.

2.     A bag contains 5 blue marbles and 3 red marbles and 6 yellow marbles. If we randomly select one marble from the bag, what is the probability of getting a blue or a red marble? Since the three outcomes are mutually exclusive, the probability of getting a blue or a red marble is the sum of their individual probabilities, which is 5/14 + 3/14 = 8/14 = 4/7.

3.     In a game of cards, we want to find the probability of drawing a heart or a spade. Since a card cannot be both a heart and a spade, the two outcomes are mutually exclusive. Therefore, the probability of drawing a heart or a spade is the sum of their individual probabilities, which is 13/52 + 13/52 = 26/52 = 1/2.

The General Rule of Addition

The General Rule of Addition, also known as the Addition Rule, is a fundamental principle in probability theory that is used to determine the probability of the union of two events. The rule applies to any two events, whether they are mutually exclusive or not.

The rule states that the probability of either one of two events occurring is the sum of the probabilities of each individual event, minus the probability of both events happening at the same time. This is because if we simply add the probabilities of both events, we would be double-counting the probability of the overlapping event.

In mathematical notation, the rule can be written as:

P(A or B) = P(A) + P(B) - P(A and B)

Where P(A)

is the probability of event A occurring, P(B) is the probability of event B occurring, and P(A and B) is the probability of both events A and B occurring simultaneously.

The General Rule of Addition Examples 

Example 1: A factory produces two types of products: Product A and Product B. The probability that a product is defective for Product A is 0.12, and the probability that a product is defective for Product B is 0.08. The probability of a product being defective for both products is 0.02. What is the probability that a randomly selected product is defective?

To solve this problem, we can use the formula: P(A or B) = P(A) + P(B) - P(A and B)

Where P(A or B) represents the probability of either event A or event B happening, P(A) represents the probability of event A happening, P(B) represents the probability of event B happening, and P(A and B) represents the probability of both events A and B happening.

In this case, we want to find the probability that a randomly selected product is defective, which is the probability of either Product A or Product B being defective. So we can set A as the event that Product A is defective, and B as the event that Product B is defective.

Using the given probabilities, we have:

P(A) = 0.12 ,

P(B) = 0.08

P(A and B) = 0.02

Substituting these values into the formula, we get:

P(A or B) = P(A) + P(B) - P(A and B)

P(A or B) = 0.12 + 0.08 - 0.02

P(A or B) = 0.18

Therefore, the probability that a randomly selected product is defective is 0.18 or 18%.

Example 2: A restaurant offers two types of desserts, cake and ice cream. The probability of a customer choosing cake is 0.7, and the probability of a customer choosing ice cream is 0.4. The probability of a customer choosing both cake and ice cream is 0.3. What is the probability that a customer will choose either cake or ice cream (or both)?

Let A be the event that a customer chooses cake, and let B be the event that a customer chooses ice cream. We want to find P(A or B), which is the probability that a customer chooses either cake or ice cream (or both).

We can start by using the formula: P(A or B) = P(A) + P(B) - P(A and B) where P(A and B) is the probability that a customer chooses both cake and ice cream.

From the problem, we know that:

- P(A) = 0.7

- P(B) = 0.4

- P(A and B) = 0.3

So, substituting these values into the formula, we get:

P(A or B) = 0.7 + 0.4 - 0.3

          = 0.8

Therefore, the probability that a customer will choose either cake or ice cream (or both) is 0.8.

The Complement Rule

The complement rule is a basic probability principle that states that the probability of an event occurring is equal to one minus the probability of that event not occurring. In other words, if the probability of event A is P(A), then the probability of not-A is 1 - P(A), and vice versa.

The complement rule is often used to find the probability of an event indirectly, by first finding the probability of its complement, and then subtracting that value from 1.

For example, if the probability of a student passing a test is 0.8, then the probability of the student failing the test is 1 - 0.8 = 0.2, because either the student passes or fails the test, and these two outcomes are mutually exclusive and collectively exhaustive.

The complement rule is a useful tool in probability theory and statistics, as it allows us to calculate probabilities indirectly and to check our calculations for consistency.

The Complement Rule Examples:

1. If the probability of rolling a 4 on a fair die is 1/6, then the probability of not rolling a 4 is 1 - 1/6 = 5/6.

2.     If the probability of a basketball player making a free throw is 0.8, then the probability of missing the free throw is 1 - 0.8 = 0.2.

3.     If the probability of a student passing a test is 0.7, then the probability of failing the test is 1 - 0.7 = 0.3.

4.     If the probability of a traffic light being green is 0.4, then the probability of the traffic light being red or yellow is 1 - 0.4 = 0.6.

5.     If the probability of an event A happening is 0.3, then the probability of event A not happening is 1 - 0.3 = 0.7.

What are Independent events?

Independent events are events where the outcome of one event does not affect the outcome of the other event. In other words, the occurrence of one event has no impact on the probability of the occurrence of the other event. For example, if you flip a coin and then roll a die, these events are independent because the outcome of the coin flip does not impact the outcome of the die roll. The probability of getting heads on the coin and rolling a 6 on the die is the product of the individual probabilities: 1/2 x 1/6 = 1/12.

Independent events Examples:

1. Tossing a coin and rolling a die: The outcome of the coin toss does not affect the outcome of the die roll.

2.     Drawing a card from a deck and then flipping a coin: The outcome of drawing a card does not affect the outcome of the coin flip.

3.     Picking a marble from a bag and then rolling a ball: The outcome of picking a marble does not affect the outcome of rolling a ball.

4.     Selecting a chocolate from a box and then drawing a card from a deck: The outcome of selecting a chocolate does not affect the outcome of drawing a card.

5.     Spinning a roulette wheel and rolling a die: The outcome of the roulette spin does not affect the outcome of the die roll.

The Special Rule of Multiplication

The special rule of multiplication is a probability rule that applies when two events are independent, meaning that the occurrence of one event does not affect the probability of the other event. It states that the probability of both events occurring is equal to the product of their individual probabilities. In mathematical notation, it is written as P(A and B) = P(A) x P(B), where P(A) is the probability of event A and P(B) is the probability of event B.

The Special Rule of Multiplication Examples:

• Flipping a coin and rolling a die. The probability of getting heads on a coin is 1/2, and the probability of rolling a 3 on a die is 1/6. The probability of getting heads and rolling a 3 is (1/2) x (1/6) = 1/12.

• Choosing a marble from a bag and then putting it back in the bag. The probability of choosing a red marble from a bag of marbles is 1/3, and the probability of choosing a blue marble is 1/3. The probability of choosing a red marble and then choosing a blue marble is (1/3) x (1/3) = 1/9.

• Rolling two dice. The probability of rolling a 1 on a die is 1/6, and the probability of rolling a 4 on a die is also 1/6. The probability of rolling a 1 on one die and a 4 on the other die is (1/6) x (1/6) = 1/36.

The General Rule of Multiplication

The general rule of multiplication states that to find the probability of two or more events occurring together, you need to multiply the probability of the first event by the probability of the second event given that the first event has occurred.

Mathematically, if A and B are two events, then the probability of both A and B occurring together is:

P(A and B) = P(A) x P(B | A)

Where P(A) is the probability of event A occurring, and P(B | A) is the conditional probability of event B occurring given that event A has occurred.

The General Rule of Multiplication  Examples:

1. Suppose you have a bag of 10 marbles, 4 of which are red and 6 of which are blue. If you randomly choose 2 marbles from the bag without replacement, what is the probability that both marbles are blue? Solution: P(Blue and Blue) = P(Blue) x P(Blue|Blue not replaced) = (6/10) x (5/9) = 0.33.

2.     A company has a team of 10 employees, 4 of whom are women and 6 of whom are men. If 3 employees are chosen at random to attend a conference, what is the probability that all 3 employees are men? Solution: P(Male and Male and Male) = P(Male) x P(Male|Male not replaced) x P(Male|Male and Male not replaced) = (6/10) x (5/9) x (4/8) = 0.1667.

3.     A restaurant has a menu of 8 appetizers and 10 main courses. If a customer selects 2 dishes at random, what is the probability that both dishes are appetizers? Solution: P(Appetizer and Appetizer) = P(Appetizer) x P(Appetizer|Appetizer not replaced) = (8/18) x (7/17) = 0.23.

What is a Contingency Table?

A contingency table is a table used to display the frequency distribution of two or more variables. It shows the number (or proportion) of individuals in each category of one variable depending on the categories of the other variable. It is often used to investigate the association between two variables.

Contingency Table Example  

Suppose we want to investigate the relationship between gender and political affiliation. We randomly sample 100 people from a population and record their gender and political affiliation. The data is summarized in the following contingency table:

|              | Democrat | Republican | Independent |

|----------|-      ---------|---       ---------|-----          --------|

| Male       |          15       |     20         |       10                |

| Female   | 25       |     10         | 20       |

In this table, the rows represent the categories of gender (male and female) and the columns represent the categories of political affiliation (Democrat, Republican, and Independent). The numbers in the cells represent the number of individuals that fall into each combination of gender and political affiliation. For example, there are 15 males who identify as Democrats, 25 females who identify as Democrats, etc.

We can use this table to calculate various statistics, such as the marginal distributions (the total number of individuals in each category) and the conditional probabilities (the probability of one variable given another variable). For example, the marginal distribution of gender is:

|              | Democrat | Republican | Independent | Total |

|-------- |------     ---|----------- -|---------- |-------|

| Male         | 15      | 20 | 10        | 45    |

| Female       | 25            | 10          | 20    | 55    |

| Total        |      40     | 30         | 30          | 100   |

We can see that there are 45 males and 55 females in the sample, and the total number of individuals is 100. Similarly, we can calculate the marginal distribution of political affiliation and the conditional probabilities of one variable given another variable.

What is Factorial?

Imagine you have three different colored balls - red, green, and blue. You want to arrange these balls in a particular order, say, from left to right. You can start by picking one of the balls for the first position - you have three choices, red, green, or blue. Once you have picked one for the first position, you have two choices left for the second position. Finally, for the third position, you only have one choice left. So, the total number of ways you can arrange the three balls in a particular order is:

3 x 2 x 1 = 6

This is an example of factorial. Factorial is denoted by the exclamation mark (!) symbol, and it tells you to multiply a certain number by all the smaller whole numbers that come before it. For example, 5! (read as "five factorial") is:

5 x 4 x 3 x 2 x 1 = 120

So, if you have five different objects and you want to arrange them in a particular order, there are 120 different ways you can do it.

Permutation Description

Imagine you have 4 different letters - A, B, C, and D. You want to know how many ways you can arrange these 4 letters in a particular order.

To start, you pick one of the letters for the first position - you have 4 choices. Once you have picked one for the first position, you have 3 choices left for the second position. For the third position, you have 2 choices left, and for the fourth position, you have 1 choice left.

So, the total number of ways you can arrange the 4 letters in a particular order is:

4 x 3 x 2 x 1 = 24

This means that there are 24 different permutations of the 4 letters A, B, C, and D.

In general, the formula for finding the number of permutations of n objects taken r at a time is:

P(n, r) = n! / (n - r)!

Where n is the total number of objects, and r is the number of objects you want to arrange in a particular order.

For example, if you have 6 different books and you want to arrange 3 of them in a particular order, the number of permutations is:

P(6, 3) = 6! / (6 - 3)! = 6! / 3! = 120 / 6 = 20

So there are 20 different ways to arrange 3 books out of 6 in a particular order.

Combination Description

Imagine you have a bag with 5 different colored balls - red, green, blue, yellow, and purple. You want to know how many different ways you can choose 2 balls from the bag, regardless of their order.

To count the number of combinations, you need to consider all the possible pairs of balls you can choose. For the first ball, you have 5 choices. For the second ball, you only have 4 choices left because you already chose one ball. But since the order doesn't matter, you need to divide by 2 to account for the fact that choosing, for example, the red ball and then the green ball is the same as choosing the green ball and then the red ball.

So, the total number of ways you can choose 2 balls from the bag, regardless of their order, is:

(5 x 4) / 2 = 10

This is an example of a combination. A combination is a way of selecting a subset of objects from a larger set, where the order in which the objects are selected doesn't matter.

The formula for finding the number of combinations of n objects taken r at a time is:

C(n, r) = n! / (r! * (n - r)!)

Where n is the total number of objects, and r is the number of objects you want to choose from the larger set.

For example, if you have 8 different books and you want to choose 3 of them to take on a trip, the number of combinations is:

C(8, 3) = 8! / (3! * (8 - 3)!) = 56

So there are 56 different ways to choose 3 books from 8, regardless of their order.