Probability and Combinations Quiz
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Questions and Answers

What is the probability of rolling a 3 on a fair six-sided die?

  • 2/3
  • 1/3
  • 1/6 (correct)
  • 1/2

What is the probability of the event 'obtaining at least one head' in the experiment of tossing a coin twice?

  • 1/4
  • 1/2
  • 3/4 (correct)
  • 1

In a standard deck of 52 cards, what is the probability of drawing a King?

  • 1/52
  • 4/52
  • 12/52
  • 1/13 (correct)

In the experiment of tossing a coin and rolling a die, what is the probability of the event 'obtaining a head and an even number'?

<p>1/6 (D)</p> Signup and view all the answers

You have a bag with 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of picking a blue marble?

<p>3/10 (A)</p> Signup and view all the answers

Which of the following is NOT a characteristic of a random experiment?

<p>The outcome is always unique. (A)</p> Signup and view all the answers

A coin is flipped 10 times. What is the probability of getting heads on all 10 flips?

<p>1/1024 (D)</p> Signup and view all the answers

Which of the following events are mutually exclusive in the experiment of tossing a coin twice? (Select all that apply)

<p>Obtaining one tail and obtaining two tails (B), Obtaining two heads and obtaining two tails (D)</p> Signup and view all the answers

A bag contains 3 red balls and 2 blue balls. Two balls are drawn at random without replacement. What is the probability that both balls are red?

<p>9/20 (D)</p> Signup and view all the answers

In a standard deck of 52 cards, what is the probability of drawing a heart, then a diamond, without replacement?

<p>1/16 (B)</p> Signup and view all the answers

In the experiment of tossing a coin and rolling a die, are the events 'obtaining a head' and 'obtaining an even number' independent?

<p>Yes, because the outcome of the coin toss does not affect the outcome of the die roll. (B)</p> Signup and view all the answers

If you roll a standard six-sided die twice, what is the probability of getting a 1 on the first roll and a 6 on the second roll?

<p>1/36 (C)</p> Signup and view all the answers

What does the formula P(A) = m/n represent?

<p>The probability of event A occurring in a sample space with 'm' favorable outcomes and a total of 'n' possible outcomes. (C)</p> Signup and view all the answers

If P(A) = 0.6, what is the probability of the event 'not A'?

<p>0.4 (A)</p> Signup and view all the answers

A bag contains 4 red balls, 3 blue balls, and 2 green balls. What is the probability of drawing a red ball, then a blue ball, without replacement?

<p>4/27 (C)</p> Signup and view all the answers

Which of the following statements is TRUE about the probability of an event?

<p>The probability of an event always lies between 0 and 1. (B)</p> Signup and view all the answers

What is the value of ⁵C₂?

<p>10 (A)</p> Signup and view all the answers

How many different ways can you choose a team of 3 students from a group of 7 students?

<p>35 (B)</p> Signup and view all the answers

A student has 8 different books. In how many ways can the student choose 3 books to take on a trip?

<p>56 (A)</p> Signup and view all the answers

If you have 6 different flavors of ice cream and you want to choose 3 flavors for a sundae, how many different combinations of ice cream can you make?

<p>20 (A)</p> Signup and view all the answers

How many different ways can you arrange the letters in the word 'PEACH'?

<p>120 (C)</p> Signup and view all the answers

A basketball team has 10 players. How many different starting lineups can they form if they need 5 players on the court?

<p>252 (A)</p> Signup and view all the answers

In how many ways can 5 people be seated in a row of 7 chairs?

<p>2520 (D)</p> Signup and view all the answers

How many different ways are there to select a president, vice president, and secretary from a club of 12 members?

<p>1320 (B)</p> Signup and view all the answers

How many ways can the word 'STATISTIC' be arranged?

<p>50,400 (D)</p> Signup and view all the answers

How many ways can 10 students be seated on 4 seats?

<p>10P4 (B), 10!/6! (C), 10x9x8x7x6/6! (D)</p> Signup and view all the answers

How many 4-digit numbers can be formed using digits 0, 1, 2, 3...9 if repetition is allowed?

<p>9x10x10x10 (B), 9000 (C)</p> Signup and view all the answers

How many 4-digit numbers can be formed using digits 0, 1, 2, 3...9 if repetition is not allowed and the last digit must be zero?

<p>9x8x7 (B)</p> Signup and view all the answers

What is the sample space when a die is tossed once?

<p>{1, 2, 3, 4, 5, 6} (B)</p> Signup and view all the answers

What is the probability of getting a sum of 4 when two dice are tossed?

<p>3/36 (B)</p> Signup and view all the answers

What is the probability of getting a sum of 6 or 7 when two dice are tossed?

<p>11/36 (D)</p> Signup and view all the answers

What is the probability of event A, given that the number of elements in event A is 'n(A)' and the number of elements in the sample space is 'n(S)'?

<p>P(A) = n(A)/n(S) (B)</p> Signup and view all the answers

Two events, A and B, are considered independent if:

<p>The occurrence of one event does not affect the probability of the other event. (A)</p> Signup and view all the answers

What is the formula for calculating the probability of the union of two events, A1 and A2, when they are not disjoint?

<p>P(A1UA2) = P(A1) + P(A2) - P(A1nA2) (A)</p> Signup and view all the answers

A bag contains 10 red balls and 5 blue balls. What is the probability of picking a red ball, given that one blue ball has already been taken out?

<p>10/14 (D)</p> Signup and view all the answers

If events A1 and A2 are disjoint, what is the formula to calculate the probability of the union of these events (A1UA2)?

<p>P(A1UA2) = P(A1) + P(A2) (A)</p> Signup and view all the answers

In a sample space 'S', an event 'A' is considered to be a subset of 'S' if:

<p>All elements of 'A' are also elements of 'S'. (D)</p> Signup and view all the answers

What is the probability of getting a head when tossing a fair coin?

<p>1/2 (A)</p> Signup and view all the answers

A die is rolled. What is the probability of getting an even number?

<p>1/2 (D)</p> Signup and view all the answers

If the probability of a contractor getting a plumbing contract is 2/5 and the probability of not getting an electric contract is 9/10, what is the probability of getting both a plumbing and electric contract if he has already got an electric contract?

<p>2/15 (B)</p> Signup and view all the answers

Flashcards

Statistical Probability

Probability based on the ratio of favorable outcomes to total outcomes in repeated experiments.

Fair Die

A die where all faces have an equal chance of landing face up (1/6 for each number).

Probability of Five on Die

The probability of rolling a five on a fair die is 1/6.

Probability Greater than Four on Die

The probability of rolling a number greater than 4 is 1/3.

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Probability of Even Number on Die

The probability of rolling an even number is 1/2.

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Combinations

Selections of items where order does not matter (e.g., AB = BA).

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Drawing Cards from a Pack

Calculating probabilities by drawing cards from a total of 52 cards based on favorable outcomes.

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Probability of Two Kings, Two Aces

The calculation involves choosing 1 king and 1 ace from 4 available each, divided by total combinations of 4 cards.

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Random Experiment

An experiment with unpredictable outcomes during each trial.

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Trial

One performance of a random experiment.

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Event

An outcome or combination of outcomes from a trial.

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Mutually Exclusive Event

Events that cannot occur at the same time.

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Independent Event

Events where the outcome of one does not affect the other.

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Probability of Event A

P(A) = m/n where m are favorable outcomes and n is total outcomes.

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Remark on Probability

P(A) is between 0 and 1 (0 ≤ P(A) ≤ 1).

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Classical Probability Limitations

Fails if n (number of outcomes) is infinite.

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Pairwise Independence

Three events A, B, and C are pairwise independent if every pair of events is independent.

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Probability of Union

The probability of A or B occurring is P(A ∪ B) = P(A) + P(B) - P(A, B).

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Conditional Probability

The probability of event A given event B is P(A | B) = P(A, B) / P(B).

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Permutations

Ordered arrangements of objects, considering the order.

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Sample Space

Set of all possible outcomes in a random experiment.

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Probability

Likelihood of an event occurring, expressed as a number between 0 and 1.

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Factorial

The product of all positive integers up to a certain number (n!).

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nPr (Permutation Formula)

Formula to find permutations: nPr = n! / (n-r)!.

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nCr (Combination Formula)

Formula to find combinations: nCr = n! / [r!(n-r)!].

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Event Probability

Probability of an event occurring from a sample space.

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Disjoint Events

Events that cannot occur at the same time.

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Joint Events

Events that can occur at the same time.

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Probability of Joint Events

P(A1 ∪ A2) = P(A1) + P(A2) - P(A1 ∩ A2).

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Multiplication Theorem

P(A1 ∩ B1) = P(A1|B1) * P(B1).

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Calculating Probability

P(A) = n(A) / n(S), where n(A) is the number of favorable outcomes.

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Combination (nCr)

Selection of r objects from n without regard to order.

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Formula for nCr

nCr = n! / (r!(n - r)!) for combinations calculation.

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Example of 4C2

Calculating combinations from 4 objects taken 2 at a time yields 6.

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Permutation (nPr)

Arrangement of r objects from n, where order matters.

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Formula for nPr

nPr = n! / (n - r)! for arrangements calculation.

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Arrangements of ABBA

The arrangements of the letters in ABBA equals 12 unique ways.

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Same gender combinations

From 3 males and 4 females, 9 sets of same-gender pairs can be formed.

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Different gender combinations

12 sets of students selected from 3 males and 4 females can be made.

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Study Notes

Probability

  • Probability is a measure of the outcome or result of an expectation (event) expressed numerically.

Permutation

  • A permutation is a selection or arrangement of different objects with order.

Combination

  • A combination is a selection or arrangement of different objects without order.

Random Experiment

  • A random experiment is one where the outcome cannot be predetermined.
  • The results of repeated experiments are essentially homogeneous.
  • The result of an experiment is unique but may be any of the possible outcomes.

Terminology

  • Trial: Performing a random experiment.
  • Event: An outcome or combination of outcomes.
  • Exclusive Cases: The total number of possible outcomes in an experiment.

Probability of an Event

  • The probability of an event is the ratio of favorable outcomes to the total possible outcomes.
  • A probability lies between 0 and 1 (inclusive).

Independent Events

  • Events are independent if the occurrence of one event does not affect the likelihood of another event happening.

Mutually Exclusive Events

  • Mutually exclusive events cannot happen at the same time.

Statistical (Empirical) Probability

  • Calculated by repeating an experiment under homogenous conditions.
  • The ratio of favorable outcomes to total trials approaches a limiting value as the number of trials becomes infinite.

Examples and Calculations

  • Examples of probability calculations using dice, coins, and cards are provided.
  • Calculations and formulas for specific probability scenarios are presented numerically.
  • Techniques for dealing with combinations and permutations are demonstrated.

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Description

Test your understanding of key concepts in probability, including permutations, combinations, and independent events. This quiz will help you reinforce your knowledge of random experiments and the fundamentals of probability theory.

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