Podcast
Questions and Answers
What is the probability of rolling a 3 on a fair six-sided die?
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?
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?
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'?
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'?
You have a bag with 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of picking a blue marble?
You have a bag with 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of picking a blue marble?
Which of the following is NOT a characteristic of a random experiment?
Which of the following is NOT a characteristic of a random experiment?
A coin is flipped 10 times. What is the probability of getting heads on all 10 flips?
A coin is flipped 10 times. What is the probability of getting heads on all 10 flips?
Which of the following events are mutually exclusive in the experiment of tossing a coin twice? (Select all that apply)
Which of the following events are mutually exclusive in the experiment of tossing a coin twice? (Select all that apply)
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?
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?
In a standard deck of 52 cards, what is the probability of drawing a heart, then a diamond, without replacement?
In a standard deck of 52 cards, what is the probability of drawing a heart, then a diamond, without replacement?
In the experiment of tossing a coin and rolling a die, are the events 'obtaining a head' and 'obtaining an even number' independent?
In the experiment of tossing a coin and rolling a die, are the events 'obtaining a head' and 'obtaining an even number' independent?
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?
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?
What does the formula P(A) = m/n represent?
What does the formula P(A) = m/n represent?
If P(A) = 0.6, what is the probability of the event 'not A'?
If P(A) = 0.6, what is the probability of the event 'not A'?
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?
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?
Which of the following statements is TRUE about the probability of an event?
Which of the following statements is TRUE about the probability of an event?
What is the value of ⁵C₂?
What is the value of ⁵C₂?
How many different ways can you choose a team of 3 students from a group of 7 students?
How many different ways can you choose a team of 3 students from a group of 7 students?
A student has 8 different books. In how many ways can the student choose 3 books to take on a trip?
A student has 8 different books. In how many ways can the student choose 3 books to take on a trip?
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?
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?
How many different ways can you arrange the letters in the word 'PEACH'?
How many different ways can you arrange the letters in the word 'PEACH'?
A basketball team has 10 players. How many different starting lineups can they form if they need 5 players on the court?
A basketball team has 10 players. How many different starting lineups can they form if they need 5 players on the court?
In how many ways can 5 people be seated in a row of 7 chairs?
In how many ways can 5 people be seated in a row of 7 chairs?
How many different ways are there to select a president, vice president, and secretary from a club of 12 members?
How many different ways are there to select a president, vice president, and secretary from a club of 12 members?
How many ways can the word 'STATISTIC' be arranged?
How many ways can the word 'STATISTIC' be arranged?
How many ways can 10 students be seated on 4 seats?
How many ways can 10 students be seated on 4 seats?
How many 4-digit numbers can be formed using digits 0, 1, 2, 3...9 if repetition is allowed?
How many 4-digit numbers can be formed using digits 0, 1, 2, 3...9 if repetition is allowed?
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?
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?
What is the sample space when a die is tossed once?
What is the sample space when a die is tossed once?
What is the probability of getting a sum of 4 when two dice are tossed?
What is the probability of getting a sum of 4 when two dice are tossed?
What is the probability of getting a sum of 6 or 7 when two dice are tossed?
What is the probability of getting a sum of 6 or 7 when two dice are tossed?
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)'?
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)'?
Two events, A and B, are considered independent if:
Two events, A and B, are considered independent if:
What is the formula for calculating the probability of the union of two events, A1 and A2, when they are not disjoint?
What is the formula for calculating the probability of the union of two events, A1 and A2, when they are not disjoint?
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?
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?
If events A1 and A2 are disjoint, what is the formula to calculate the probability of the union of these events (A1UA2)?
If events A1 and A2 are disjoint, what is the formula to calculate the probability of the union of these events (A1UA2)?
In a sample space 'S', an event 'A' is considered to be a subset of 'S' if:
In a sample space 'S', an event 'A' is considered to be a subset of 'S' if:
What is the probability of getting a head when tossing a fair coin?
What is the probability of getting a head when tossing a fair coin?
A die is rolled. What is the probability of getting an even number?
A die is rolled. What is the probability of getting an even number?
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?
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?
Flashcards
Statistical Probability
Statistical Probability
Probability based on the ratio of favorable outcomes to total outcomes in repeated experiments.
Fair Die
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
Probability of Five on Die
The probability of rolling a five on a fair die is 1/6.
Probability Greater than Four on Die
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
Probability of Even Number on Die
The probability of rolling an even number is 1/2.
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Combinations
Combinations
Selections of items where order does not matter (e.g., AB = BA).
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Drawing Cards from a Pack
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
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
Random Experiment
An experiment with unpredictable outcomes during each trial.
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Trial
Trial
One performance of a random experiment.
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Event
Event
An outcome or combination of outcomes from a trial.
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Mutually Exclusive Event
Mutually Exclusive Event
Events that cannot occur at the same time.
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Independent Event
Independent Event
Events where the outcome of one does not affect the other.
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Probability of Event A
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
Remark on Probability
P(A) is between 0 and 1 (0 ≤ P(A) ≤ 1).
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Classical Probability Limitations
Classical Probability Limitations
Fails if n (number of outcomes) is infinite.
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Pairwise Independence
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
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
Conditional Probability
The probability of event A given event B is P(A | B) = P(A, B) / P(B).
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Permutations
Permutations
Ordered arrangements of objects, considering the order.
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Sample Space
Sample Space
Set of all possible outcomes in a random experiment.
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Probability
Probability
Likelihood of an event occurring, expressed as a number between 0 and 1.
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Factorial
Factorial
The product of all positive integers up to a certain number (n!).
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nPr (Permutation Formula)
nPr (Permutation Formula)
Formula to find permutations: nPr = n! / (n-r)!.
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nCr (Combination Formula)
nCr (Combination Formula)
Formula to find combinations: nCr = n! / [r!(n-r)!].
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Event Probability
Event Probability
Probability of an event occurring from a sample space.
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Disjoint Events
Disjoint Events
Events that cannot occur at the same time.
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Joint Events
Joint Events
Events that can occur at the same time.
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Probability of Joint Events
Probability of Joint Events
P(A1 ∪ A2) = P(A1) + P(A2) - P(A1 ∩ A2).
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Multiplication Theorem
Multiplication Theorem
P(A1 ∩ B1) = P(A1|B1) * P(B1).
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Calculating Probability
Calculating Probability
P(A) = n(A) / n(S), where n(A) is the number of favorable outcomes.
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Combination (nCr)
Combination (nCr)
Selection of r objects from n without regard to order.
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Formula for nCr
Formula for nCr
nCr = n! / (r!(n - r)!) for combinations calculation.
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Example of 4C2
Example of 4C2
Calculating combinations from 4 objects taken 2 at a time yields 6.
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Permutation (nPr)
Permutation (nPr)
Arrangement of r objects from n, where order matters.
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Formula for nPr
Formula for nPr
nPr = n! / (n - r)! for arrangements calculation.
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Arrangements of ABBA
Arrangements of ABBA
The arrangements of the letters in ABBA equals 12 unique ways.
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Same gender combinations
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
Different gender combinations
12 sets of students selected from 3 males and 4 females can be made.
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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.