Probability

The counting techniques in probability, statistics, mathematics, engineering, and computer science are essential tools. Counting Techniques in probability help in determining the number of ways a particular event can occur.

The following are the most common counting techniques in probability theory:

Factorial

For any integer $n$, $n$ factorial (denoted by $n!$) is the descending product beginning with $n$ and ending with 1. It can be written as

$$n! = n\times (n-1) \times (n-2) \times \cdots \times 2 \times 1$$

The example of factorial counting are:

• $3! = 3\times 2\times 1 = 6$
• $5! = 5\times 4\times 3! = 20 \times 6 = 120$
• $10! = 10\times 9\times 8\times 7\times 6\times 5! = 3628800$

Note that a special definition is made for the case of $0!$, $0!=1$.

Permutations

A permutation of a group of objects is an ordered arrangement of the objects. The number of different permutations of a group of $n$ objects is $n!$. The formula of permutation is

$$P(n, r) = {}^nP_r = \binom{n}{r} = \frac{n!}{(n-r)!}$$

where $n$ is the total number of objects, and $r$ is the number of objects to be arranged.

The example of permutations are:

• The number of ways of dealing with the cards of a standard deck in some order is $52! = 8.066\times 10^{67}$
• Suppose, we want to place a set of five names in some order, there are five choices for which name to place first, then 4 choices of which to list second, 3 choices for third, 2 choices for fourth, and only one choice for the last (fifth one). Therefore, one can determine, how many different ways can 5 people be ordered in a row can be counted using the fundamental counting principle, the number of different ways to put 5 names in order is $5! = 5\times 4 \times 3\times 2\times 1 = 120$

Often entire set of objects is not required to be placed in order, usually one wants to compute how many ways a few chosen objects can be ordered. For example,

Example: A horse race has 14 horses, how many different possible ways can the top 3 horses finish?
Solution: There are 14 possibilities for which horse finishes first, 13 for second, and 12 for third. So, by the fundamental counting principle, there are $14\times 13\times 12 = 2184$ different possible ways (finishing orders) for the top three horses. $\binom{14}{3} = \frac{14!}{(14-3)!}=2184$.

In the above examples, permutations are called permutations of $n$ objects taken $r$ at a time.

Combinations

A combination is a selection of objects from a set without regard to order. Combinations are used when calculating the number of outcomes for experiments involving multiple choices, and often the order of the choices is not required. The formula of combination is

$$C(n, r) = {}^nC_r = \frac{n!}{r!(n-r)!}=\frac{{}^nP_r}{r!}$$

where $n$ is the total number of objects, and $r$ is the number of objects to be arranged without any regard to importance or order.

The example of combinations are:

• Drawing a 5-card poker hand (${}^{52}C_5$)
• Selecting a three-person committee from a group of 30 (without any priority or importance) (${}^{30}C_3$)

A choice of $r$ objects from a group of $n$ objects without regard to order is called a combination of $n$ objects taken $r$ at a time.

Example: In how many different ways can a committee of 3 people be chosen from a group of 10 people?

Solution: $C(10, 3) = \frac{10!}{3!(10-3)!} = 120 ways$

Multiplication Principle

If one event can occur $m$ times and another event occurs $n$ times, then the occurrence of the two events together can be computed using the multiplication principle, that is, by multiplying $m\times n$. For example, if there are 5 shirts and 3 pants to choose from, one can compute the different ways of outfits by multiplying the number of shirts and number of pants, i.e., $5 \times 3=15$, so there are 15 ways of outfits from 5 shirts and 3 pants.

Addition Principle

If one event can occur in $m$ ways and a second event can occur n $n$ ways, then one or the other event can occur in $m+n$ ways. For example, if there are 3 red balls and 4 blue balls, the number of was a ball can be chosen is: $4+3=7$.

Application of Counting Techniques in Probability

• Probability: Calculating probabilities of events based on the number of favorable outcomes and the total number of possible outcomes.
• Combinatorics: Studying the arrangement, combination, or selection of objects.
• Computer Science: Analyzing algorithms and data structures.
• Statistics: Sampling and hypothesis testing.
• Cryptography: Designing secure encryption methods

FAQs about Counting Techniques in Probability

1. What is meant by counting techniques?
2. What are the applications of counting techniques in probability?
3. Define permutations and combinations.
4. What is the difference between the multiplication and addition principles?
5. Give real-life examples of permutations and combinations.
6. Write down the formulas of permutations and combinations.

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The post is about the Probability MCQS Online Test. There are 20 multiple-choice questions covering the basics of probability, events and types of events, rules of probability, subjective and classical probability, etc. Let us start with the Probability MCQs Online Test.

Probability MCQs Online Test

• A die is rolled. What is the probability that the number rolled is greater than 2 and even
• If you have two standard decks of cards (one with a blue back and the other with a red back), you draw one card from each. What is the probability that the blue-backed card is a heart or the red-backed card is a black card?
• If you roll a pair of dice, what is the probability that (at least) one dice is a 1 or the sum of the dice is 9?
• If a card is chosen from a standard deck of cards, what is the probability of getting a diamond or a club?
• If a card is chosen from a standard deck of cards, what is the probability of getting a two or a five?
• If a card is chosen from a standard deck of cards, what is the probability of getting a five or a face card?
• If a letter is chosen at random from the 10 letters of the word STATISTICS, what is the probability that it is a vowel?
• From the following table, what is the probability of selecting a female university graduate student from this group?
• Subjective probabilities are assigned to the events $A$ and $B$, which comprise a sample space. Which of the following probability statements is not valid?
• A personal manager selects an applicant at random from a large group for an interview. The probability of the applicant being male is 0.60. The probability of selecting an adult is 0.70. The probability of selecting an adult male is 0.45. Given that a male is selected, the probability that he is an adult is:
• Which of the following is a collection of all mutually exclusive events representing a card randomly selected from a deck of ordinary playing cards?
• Which of the following are collectively exhaustive events representing a card randomly selected from a deck of ordinary playing cards?
• Indicate in which one of the following situations the events $A$ and $B$ are independent:
• Which one of the following statements is false?
• A marginal probability might be found by any but which one of the following?
• Which one of the following statements is not true?
• If $A$ and $B$ are independent events, $P(A) = 0.45, P(B) = 0.60$ then $P(A \cap B)$ is
• The probability of drawing one white ball from a bag containing 2 white, 3 blue, and 3 black balls is
• A letter is chosen at random from the word MATHEMATICS, the probability of getting $M$ is
• If $P(B|A) = P(B)$ then $A$ and $B$ are

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The post is about MCQs Probability Statistics. There are 20 multiple-choice questions covering topics related to events and types of events, basics of probability and types of probability, and addition and multiplication rules of probability. Let us start with the MCQs Probability Statistics.

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Online MCQs Probability Statistics

• Two events $A$ and $B$ are independent if and only if
• If $A$ and $B$ are mutually exclusive, then
• If the events $B_1, B_2, \cdots, B_k$ partition of this sample space $S$ that $P(B_i)\ne 0$ for $i = 1, 2, \cdots, k$)  then for any event $A$ of $S$
• Let $A_1, A_2, \cdots, A_n$ be $n$ events in an event space. If
  $P(A_iA_j) = P(A_i)P(A_j) \quad for \quad i\ne j$
  $P(A_iA_jA_k) =P(A_i)P(A_j)P(A_k) \quad for \quad i\ne j \ne k$
  $\vdots$
  $P(\cap_{i=1}^n A_i) = \prod_{i=1}^n P(A_i)$ then the events are called
• The classical probability method is applied to an experiment that
• The joint probability of two independent events $A$ and $B$ is
• Two mutually exclusive events
• The probability can never be
• The probability of an impossible event is always
• If $P(A \cap B) = 0.12$ and $P(A) = 0.3$, find $P(B)$ where $A$ and $B$ are independent
• For two mutually exclusive events $A$ and $B$, $P (A) = 0.2$ and $P (B) = 0.4$, then $P(A \cup B)$ is
• A standard deck of 52 cards is shuffled. What is the probability of choosing the 5 diamonds,
• When two coins are tossed the probability of at least one head is
• Two cards are drawn at random from a standard deck of 52 cards, without replacement. What is the probability of drawing a 7 and a king in that order?
• $P(A\cap B)=P(A)\cdot P(B)$, then $A$ and $B$ are
• To calculate posterior probability, a data professional can use ———- to update the prior probability based on the data.
• When three dice are rolled, the sample space consists of
• An event that contains the finite number point, the sample space is called
• The total area under the curve in the probability of density function is?
• If $A$ denotes the males of a town and $B$ denotes the females of that town, then $A$ and $B$ are ——-?

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The post is about the Probability MCQs Quiz. There are 25 multiple-choice questions covering the topic related to counting rules of probability, random experiments, assigning probability, events and types of events, and rules of probability. Let us start with the Probability MCQs Quiz.

Please go to Best Probability MCQs Quiz 4 to view the test

Online Probability MCQs Quiz

• A lottery is conducted using 3 urns. Each urn contains balls numbered from 0 to 9. One ball is randomly selected from each urn. The total number of sample points in the sample space is
• Three applications for admission to a university are checked to determine whether each applicant is male or female. The number of sample points in this experiment is
• Suppose your favorite cricket team has 2 games left to finish the series. The outcome of each game can be won, lost, or tied. The number of possible outcomes is
• Each customer entering a departmental store will either buy or not buy a certain product. An experiment consists of the following 3 customers and determining whether or not they will buy any certain product. The number of sample points in this experiment is as follows:
• Two letters are to be selected at random from five letters (A, B, C, D, and E). How many possible selections are there?
• The “Top Three” at a racetrack consists of picking the correct order of the first three horses in a race. If there are 10 horses in a particular race, how many “Top Three” outcomes are there?
• When the assumption of equally likely outcomes is used to assign probability values, the method used to assign probabilities is referred to as the
• A method of assigning probabilities that assumes the experimental outcomes are equally likely is called
• When the results of historical data or experimentation are used to assign probability values, the method used to assign probabilities is referred to as the
• The probability assigned to each experimental outcome must be
• An experiment consists of four outcomes with $P(A) = 0.2, P(B) = 0.3, P(C) = 0.4$. The probability of the outcome $P(D)$ is
• Given that event $A$ has a probability of 0.25, the probability of the complement of event $A$
• The symbol $\cup$ shows the
• The union of events $A$ and $B$ is the event containing
• The probability of the union of two events with non-zero probabilities
• The symbol $\cap$ shows the
• The addition law helps to compute the probabilities of
• If $P(A) = 0.38, P(B) = 0.83$, and $P(A\cap B)=0.57$, then $P(A\cup B) =$ ?
• If $P(A) = 0.62, P(B) = 0.47$, and $P(A\cup B) = 0.88$, then $P(A \cap B) =$ ?
• Two events are mutually exclusive if
• Events that have no sample points in common are called
• The probability of the intersection of two mutually exclusive events
• If two events are mutually exclusive, then the probability of their intersection
• Two events, $A$ and $B$ are mutually exclusive and each has a non-zero probability. If event $A$ is known to occur, the probability of the occurrence of event $B$ is
• If $A$ and $B$ are mutually exclusive events with $P(A)=0.3$ and $P(B)=0.5$, then $P(A \cap B)=$?

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