PRC-2 · Chapter 11
Probability Concepts MCQs with Answers
60 multiple-choice questions on Probability Concepts for PRC-2 Quantitative Analysis for Business. Try each one before revealing the answer and explanation.
Practise this chapter interactivelyQuestion 1
If events A and B are strictly mutually exclusive, and P(A) = 0.5 and P(B) = 0.4, what is the probability of either A or B occurring, denoted as P(A or B)?
- A) 0.1
- B) 0.2
- C) 0.9
- D) 0.54
Show answer & explanation
Answer: C) 0.9
Because the events are mutually exclusive and cannot happen at the same time, the probability of A or B occurring is simply the sum of their individual probabilities: 0.5 + 0.4 = 0.9.
Question 2
A 4-digit pin code can begin with any number except 0 and 1. If repetition of the same digit is allowed, what is the exact probability that a randomly generated pin code begins with a 5 and ends with a 2?
- A) 1/49
- B) 1/70
- C) 1/80
- D) 1/90
Show answer & explanation
Answer: C) 1/80
The first digit has 8 valid options (2 through 9), so the probability of starting with 5 is 1/8. The final digit has 10 valid options (0-9), so the probability of ending with 2 is 1/10. Combined: 1/8 * 1/10 = 1/80.
Question 3
In an athletic race, eight runners have an equal chance of winning. What is the mathematical probability that a spectator correctly guesses the top 3 runners in the exact correct order?
- A) 1/112
- B) 1/336
- C) 0.61%
- D) 1/512
Show answer & explanation
Answer: B) 1/336
This requires calculating permutations. The number of ways to arrange 3 winners out of 8 is 8 * 7 * 6 = 336. The probability of guessing the single correct sequence is 1/336.
Question 4
If a fair, standard six-sided dice is rolled 5 consecutive times, what is the approximate probability of rolling a '3' at least one time?
- A) 0.388
- B) 0.402
- C) 0.598
- D) 0.612
Show answer & explanation
Answer: C) 0.598
The probability of NOT rolling a 3 on a single roll is 5/6. For 5 rolls, it is (5/6)^5 = 0.4018. The probability of rolling at least one 3 is 1 - 0.4018 = 0.598.
Question 5
Two distinct events, A and B, are defined as non-mutually exclusive. If their probabilities are P(A) = 0.5 and P(B) = 0.4, and they are independent, what is P(A and B)?
- A) 0.1
- B) 0.2
- C) 0.54
- D) 0.9
Show answer & explanation
Answer: B) 0.2
For non-mutually exclusive independent events, the probability of both occurring together is found by multiplying their probabilities: P(A) * P(B) = 0.5 * 0.4 = 0.2.
Question 6
What is the collective term used to describe the entire set of all possible outcomes for a random experiment?
- A) Rejection region
- B) Sample space
- C) Null hypothesis
- D) Progressive series
Show answer & explanation
Answer: B) Sample space
In standard probability theory, the sample space represents the complete, exhaustive list of every single possible outcome that can result from an experiment.
Question 7
If a factory machine has a 0.02 probability of breaking down on any given day, what is the complementary probability that it operates perfectly?
- A) 0.02
- B) 0.50
- C) 0.98
- D) 1.00
Show answer & explanation
Answer: C) 0.98
Complementary probabilities must always sum to exactly 1. Therefore, 1 - 0.02 = 0.98 probability of perfect operation.
Question 8
Which mathematical rule is applied to find the probability of Event A 'OR' Event B occurring when the events are mutually exclusive?
- A) Multiplication Rule
- B) Addition Rule
- C) Subtraction Rule
- D) Permutation Rule
Show answer & explanation
Answer: B) Addition Rule
The Addition Rule states that for mutually exclusive events, you simply add the probabilities together: P(A or B) = P(A) + P(B).
Question 9
When calculating the probability of independent events occurring in sequence (Event A 'AND' Event B), which mathematical operation is strictly required?
- A) Addition
- B) Subtraction
- C) Multiplication
- D) Division
Show answer & explanation
Answer: C) Multiplication
To find the probability of multiple independent events happening together, you must multiply their individual probabilities: P(A) * P(B).
Question 10
What distinguishes a 'Permutation' from a 'Combination' in probability calculations?
- A) Permutations are only used for dice rolls.
- B) In combinations, the exact order of the elements is strictly important.
- C) In permutations, the exact sequence or order of the elements is strictly important.
- D) Permutations allow for infinite probabilities.
Show answer & explanation
Answer: C) In permutations, the exact sequence or order of the elements is strictly important.
Permutations calculate arrangements where the specific order matters (like guessing the top 3 runners in exact order), whereas combinations ignore the sequence.
Question 11
A 4-digit pin code can begin with any number except 0, 1, and 2. If repetition is allowed, what is the probability the pin code begins with a 5 and ends with a 3?
- A) 1/49
- B) 1/70
- C) 1/80
- D) 1/90
Show answer & explanation
Answer: B) 1/70
The first digit has 7 valid options (3 through 9), so starting with 5 has a 1/7 chance. The last digit has 10 options (0-9), so ending with 3 has a 1/10 chance. 1/7 * 1/10 = 1/70.
Question 12
If an investment scenario guarantees a payout with absolutely zero risk of failure, the mathematical probability of receiving the payout is precisely:
- A) 0
- B) 0.5
- C) 1
- D) Infinite
Show answer & explanation
Answer: C) 1
In probability theory, an event that is absolutely certain to occur always has a mathematical probability of exactly 1.
Question 13
What is the theoretical probability of an event that is physically impossible to occur?
- A) -1
- B) 0
- C) 0.5
- D) 1
Show answer & explanation
Answer: B) 0
Probability scales strictly from 0 to 1. An event that is totally impossible has a probability of precisely 0. It cannot be negative.
Question 14
When calculating expected values in a business scenario, how is the Expected Value (EV) mathematically derived?
- A) By dividing the highest possible outcome by the lowest.
- B) By multiplying each possible financial outcome by its respective probability of occurring, and summing the results.
- C) By subtracting the cost of capital from the probability.
- D) By squaring the correlation coefficient.
Show answer & explanation
Answer: B) By multiplying each possible financial outcome by its respective probability of occurring, and summing the results.
The expected value represents the weighted average of all possible outcomes, calculated by summing the products of each outcome and its associated probability.
Question 15
In standard notation, what does the expression P(A|B) technically represent?
- A) The probability of Event A minus Event B.
- B) The probability of Event A or Event B.
- C) The conditional probability of Event A occurring, given that Event B has already occurred.
- D) The probability of Event A multiplied by Event B.
Show answer & explanation
Answer: C) The conditional probability of Event A occurring, given that Event B has already occurred.
The vertical bar '|' is the standard statistical notation for conditional probability, indicating the probability of the first event happening under the strict condition that the second event is true.
Question 16
Which of the following is a mandatory mathematical requirement for any valid discrete probability distribution?
- A) The individual probabilities must be negative.
- B) The sum of all individual probabilities across the sample space must strictly equal 1.
- C) The mean must be exactly zero.
- D) Every outcome must have the exact same identical probability.
Show answer & explanation
Answer: B) The sum of all individual probabilities across the sample space must strictly equal 1.
A core axiom of probability distributions is that the sum of the probabilities of all mutually exclusive and collectively exhaustive outcomes must equal exactly 1.0.
Question 17
A discrete probability distribution describes outcomes that are:
- A) Measurable on an infinite scale
- B) Only able to take certain specific countable values
- C) Always negative in value
- D) Distributed without any gaps
Show answer & explanation
Answer: B) Only able to take certain specific countable values
Unlike continuous distributions, discrete distributions deal with variables that have distinct, separate, and countable outcomes (like the number of cars sold).
Question 18
In the context of a discrete probability distribution, the 'Expected Value' (E[X]) is mathematically synonymous with:
- A) The arithmetic mean of the distribution
- B) The standard deviation of the population
- C) The statistical mode
- D) The range between highest and lowest
Show answer & explanation
Answer: A) The arithmetic mean of the distribution
The Expected Value is the weighted average of all possible values, which represents the theoretical long-term average (mean) of the random variable.
Question 19
Which specific probability distribution is strictly characterized by having only two possible outcomes per trial (e.g., success or failure)?
- A) Normal Distribution
- B) Poison Distribution
- C) Binomial Distribution
- D) Uniform Distribution
Show answer & explanation
Answer: C) Binomial Distribution
The Binomial distribution models the number of successes in a fixed number of independent trials, where each individual trial has only two possible outcomes.
Question 20
Which of the following describes the shape of a perfectly normal distribution curve?
- A) J-shaped and rising
- B) U-shaped and dipping
- C) Bell-shaped and symmetrical
- D) Rectangular and flat
Show answer & explanation
Answer: C) Bell-shaped and symmetrical
The Normal (Gaussian) distribution is well-known for its classic symmetrical bell shape, where the mean, median, and mode all overlap at the center peak.
Question 21
In a perfectly symmetrical Normal Distribution, what specific percentage of observations theoretically fall within ±1 Standard Deviation of the mean?
- A) Approximately 50%
- B) Approximately 68%
- C) Approximately 95%
- D) Approximately 99.7%
Show answer & explanation
Answer: B) Approximately 68%
According to the Empirical Rule (68-95-99.7 rule), about 68% of the area under a normal curve is located within one standard deviation of the average.
Question 22
What are the specific parameters (Mean and Standard Deviation) of a 'Standard Normal Distribution'?
- A) Mean = 1, SD = 0
- B) Mean = 0, SD = 1
- C) Mean = 100, SD = 15
- D) Mean = 0, SD = 0
Show answer & explanation
Answer: B) Mean = 0, SD = 1
The Standard Normal Distribution (Z-distribution) is a specialized version of the normal distribution where the mean is exactly 0 and the standard deviation is exactly 1.
Question 23
If a z-score is calculated to be +2.0, what does this mathematically imply about that specific data point?
- A) The point is exactly equal to the mean.
- B) The point is exactly two units greater than the mean.
- C) The point is exactly two standard deviations above the mean.
- D) The point has a 2% probability of occurring.
Show answer & explanation
Answer: C) The point is exactly two standard deviations above the mean.
The z-score measures how many standard deviations a raw data point is located away from the mean. A positive 2.0 indicates it is two standard deviations above it.
Question 24
Which distribution is most appropriate for modeling rare, independent events that occur within a specific fixed interval of time or of space?
- A) Binomial Distribution
- B) Normal Distribution
- C) Poisson Distribution
- D) Symmetrical Distribution
Show answer & explanation
Answer: C) Poisson Distribution
The Poisson distribution is uniquely used to calculate the probability of a specific number of independent events occurring within a fixed timeframe or area.
Question 25
The total area located beneath the curve of any valid continuous probability density function (like the normal curve) is always:
- A) 0
- B) 0.5
- C) 1.0
- D) Infinite
Show answer & explanation
Answer: C) 1.0
Mirroring the rule for discrete distributions, the total area under a continuous distribution curve represents the total probability of all outcomes, which must sum to exactly 1.0.
Question 26
In a binomial experiment with n = 10 trials and a success probability p = 0.5, what is the calculated expected value (mean)?
- A) 2
- B) 5
- C) 10
- D) 1
Show answer & explanation
Answer: B) 5
The mean of a binomial distribution is calculated simply as n * p. Here, 10 * 0.5 = 5.
Question 27
Continuous probability distributions (like the Normal distribution) differ from discrete ones because they:
- A) Deal with variables that can take any value in an interval.
- B) Only deal with whole items.
- C) Have no mean or standard deviation.
- D) Are always skewed to the left.
Show answer & explanation
Answer: A) Deal with variables that can take any value in an interval.
Continuity implies that there are no gaps between possible values; a continuous variable can be measured with infinite precision within its defined range.
Question 28
If a dataset has a mean of 50 and a standard deviation of 5, what is the z-score for a data value of 40?
- A) +2.0
- B) -2.0
- C) +10.0
- D) -10.0
Show answer & explanation
Answer: B) -2.0
The formula for a z-score is z = (X - Mean) / SD. Here, (40 - 50) / 5 = -10 / 5 = -2.0.
Question 29
In a Poison Distribution, if the average number of events per hour (λ) is 4, what is the variance of the distribution?
- A) 2
- B) 4
- C) 8
- D) 16
Show answer & explanation
Answer: B) 4
A unique statistical property of the Poisson distribution is that its mean and its variance are mathematically identical (μ = σ^2 = λ).
Question 30
According to the normal distribution table, what is the area to the left of a z-score of 0?
- A) 0%
- B) 50%
- C) 100%
- D) 95%
Show answer & explanation
Answer: B) 50%
Because the Normal Distribution is perfectly symmetrical and centered at a mean (z) of 0, exactly half (50%) of the outcomes fall to the left of the center.
Question 31
A digital vault uses a 5-digit PIN code. The code can begin with any number except 0, 1, 2, and 3. If repetition of digits is strictly allowed, what is the exact probability that a randomly generated PIN begins with an 8 and ends with a 4?
- A) 1/40
- B) 1/50
- C) 1/60
- D) 1/100
Show answer & explanation
Answer: C) 1/60
The first digit has 6 valid options (4 through 9), so the probability of starting with an 8 is 1/6. The last digit has 10 valid options (0-9), so the probability of ending with a 4 is 1/10. Multiplying the independent probabilities: 1/6 * 1/10 = 1/60.
Question 32
In a corporate tournament, 5 athletes have an absolutely equal chance of winning. What is the theoretical mathematical probability that a spectator accurately predicts the top 3 finishers in their exact sequential order?
- A) 1/10
- B) 1/15
- C) 1/60
- D) 1/120
Show answer & explanation
Answer: C) 1/60
Because exact order matters, this is a permutation. The number of ways to arrange 3 winners out of 5 is 5 * 4 * 3 = 60. The probability of guessing that single specific sequence is exactly 1/60.
Question 33
If Event A and Event B are defined as strictly mutually exclusive, and P(A) = 0.35 and P(B) = 0.45, what is the probability of P(A or B)?
- A) 0.10
- B) 0.1575
- C) 0.80
- D) 1.00
Show answer & explanation
Answer: C) 0.80
For mutually exclusive events (they cannot physically happen at the same time), the Addition Rule simplifies to P(A or B) = P(A) + P(B). Therefore, 0.35 + 0.45 = 0.80.
Question 34
Events X and Y are non-mutually exclusive. If P(X) = 0.60, P(Y) = 0.50, and the probability of both occurring together P(X and Y) = 0.30, what is the probability of either X or Y occurring?
- A) 0.30
- B) 0.80
- C) 1.10
- D) 1.40
Show answer & explanation
Answer: B) 0.80
For non-mutually exclusive events, you must subtract the overlapping probability to avoid double-counting. P(X or Y) = P(X) + P(Y) - P(X and Y). 0.60 + 0.50 - 0.30 = 0.80.
Question 35
A standard, fair six-sided die is rolled exactly 4 times in sequence. What is the approximate probability of rolling a '6' at least one time during these 4 rolls?
- A) 0.166
- B) 0.482
- C) 0.518
- D) 0.666
Show answer & explanation
Answer: C) 0.518
The easiest method is to find the probability of NEVER rolling a 6 and subtracting it from 1. P(not 6) = 5/6. For 4 rolls, (5/6)^4 ≈ 0.482. The complement is 1 - 0.482 = 0.518.
Question 36
If an experiment results in an event having a theoretical probability of exactly 1.0, what does this mathematically signify?
- A) The event is absolutely impossible.
- B) The event has a 50/50 chance of occurring.
- C) The event is absolutely certain to occur.
- D) The event is mutually exclusive.
Show answer & explanation
Answer: C) The event is absolutely certain to occur.
In standard probability theory, a value of 0 means physical impossibility, while a value of exactly 1 represents absolute mathematical certainty that the event will happen.
Question 37
An investor evaluates a stock with a 20% chance of yielding a Rs. 100,000 profit and an 80% chance of yielding exactly Rs. 0. What is the mathematical 'Expected Value' of this investment?
- A) Rs. 10,000
- B) Rs. 20,000
- C) Rs. 50,000
- D) Rs. 100,000
Show answer & explanation
Answer: B) Rs. 20,000
Expected Value is calculated by multiplying each outcome by its respective probability and summing them. (0.20 * 100,000) + (0.80 * 0) = 20,000 + 0 = Rs. 20,000.
Question 38
If you draw one single card at random from a thoroughly shuffled, standard 52-card deck, what is the exact probability of drawing any 'face card' (Jack, Queen, or King)?
- A) 1/13
- B) 3/13
- C) 4/13
- D) 1/4
Show answer & explanation
Answer: B) 3/13
A standard deck contains 12 face cards in total (3 per suit * 4 suits). The probability is 12 out of 52, which simplifies mathematically to 3/13.
Question 39
Two completely independent events, C and D, have probabilities of P(C) = 0.70 and P(D) = 0.40. What is the mathematical probability of both occurring together, P(C and D)?
- A) 0.28
- B) 0.30
- C) 1.10
- D) 0.10
Show answer & explanation
Answer: A) 0.28
If events are strictly independent (one does not affect the other), the probability of both occurring is found by multiplying them: P(C) * P(D) = 0.70 * 0.40 = 0.28.
Question 40
A box contains 3 red marbles, 5 blue marbles, and 2 green marbles. If one marble is drawn at random, what is the precise probability that it is NOT blue?
- A) 0.20
- B) 0.30
- C) 0.50
- D) 0.80
Show answer & explanation
Answer: C) 0.50
There are 10 marbles total. There are 5 blue marbles, meaning there are 5 marbles that are NOT blue (3 red + 2 green). The probability is 5/10, which reduces to 0.50.
Question 41
What does the statistical term 'Sample Space' formally refer to in a probability experiment?
- A) The exact geometric center of the data.
- B) A designated subset of preferred outcomes.
- C) The complete, exhaustive set of all mathematically possible outcomes.
- D) The probability of a Type I error.
Show answer & explanation
Answer: C) The complete, exhaustive set of all mathematically possible outcomes.
The sample space (usually denoted by 'S') is mathematically defined as the complete set encompassing every single possible outcome that could occur in a specific probability experiment.
Question 42
When calculating a conditional probability denoted as P(A | B), what is the formula generally utilized?
- A) P(A) * P(B)
- B) P(A and B) / P(B)
- C) P(A) + P(B) - P(A and B)
- D) P(A) / P(A and B)
Show answer & explanation
Answer: B) P(A and B) / P(B)
The conditional probability of A given that B has already occurred is calculated by taking the probability of the intersection (where both occur) and dividing it by the probability of the given condition (B).
Question 43
Two standard, fair 6-sided dice are rolled simultaneously. What is the mathematical probability that the sum of the two upward faces is exactly 7?
- A) 1/36
- B) 1/12
- C) 1/6
- D) 1/4
Show answer & explanation
Answer: C) 1/6
There are 36 total possible combinations. The combinations that sum to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). That is 6 successful outcomes out of 36, which simplifies to 1/6.
Question 44
Which of the following scenarios strictly requires the use of 'Combinations' rather than 'Permutations'?
- A) Selecting a 4-digit password for a phone.
- B) Choosing a 1st, 2nd, and 3rd place winner in a race.
- C) Selecting a generic 3-person committee from a pool of 10 employees.
- D) Arranging 5 distinct books on a library shelf.
Show answer & explanation
Answer: C) Selecting a generic 3-person committee from a pool of 10 employees.
Combinations are mathematically used when the specific order or sequence of the selection absolutely does not matter (like a generic group of people), whereas permutations are used when order is critical.
Question 45
What is the mathematical sum of the probabilities of all mutually exclusive and exhaustive events in a given sample space?
- A) 0
- B) 0.5
- C) 1.0
- D) Infinity
Show answer & explanation
Answer: C) 1.0
A fundamental axiom of modern probability theory dictates that the sum of the probabilities of all possible, non-overlapping outcomes in an exhaustive sample space must exactly equal 1.0 (or 100%).
Question 46
What is the mathematical range of a probability value (P) for any single, specific event?
- A) -1 to +1
- B) 0 to 100
- C) 0 to 1
- D) Undefined
Show answer & explanation
Answer: C) 0 to 1
Probability is strictly bounded between 0 (representing an impossible event) and 1 (representing a mathematically certain event).
Question 47
If two events, A and B, are defined as 'mutually exclusive', what is the probability of both events occurring simultaneously P(A ∩ B)?
- A) 1
- B) 0.5
- C) Exactly 0
- D) P(A) * P(B)
Show answer & explanation
Answer: C) Exactly 0
Mutually exclusive events are events that cannot possibly happen at the same time (e.g., flipping a single coin and getting both heads and tails). Therefore, their joint probability is strictly zero.
Question 48
The 'Complementary Rule' in probability theory states that the probability of an event NOT occurring [P(A')] is equal to:
- A) P(A) / 2
- B) 1 - P(A)
- C) 1 / P(A)
- D) 0
Show answer & explanation
Answer: B) 1 - P(A)
Since the sum of all possible outcomes must equal 1, the probability of an event not happening is simply 1 minus the probability that it does happen.
Question 49
If two events are 'independent', the probability of both occurring P(A ∩ B) is calculated using the Multiplication Rule as:
- A) P(A) + P(B)
- B) P(A) - P(B)
- C) P(A) × P(B)
- D) P(A) / P(B)
Show answer & explanation
Answer: C) P(A) × P(B)
For independent events (where the outcome of one does not affect the other), the joint probability is find by multiplying the individual probabilities of each event.
Question 50
In a standard Normal Distribution, exactly what percentage of the data observations fall within ±1 standard deviation of the mean?
- A) 50%
- B) 68.27%
- C) 95.45%
- D) 99.73%
Show answer & explanation
Answer: B) 68.27%
According to the Empirical Rule, roughly 68.27% of all observations in a perfectly bell-shaped normal distribution fall within one standard deviation of the arithmetic mean.
Question 51
A 'Standard Normal Distribution' (Z-distribution) is uniquely defined as having which specific mean and standard deviation?
- A) Mean = 1, SD = 0
- B) Mean = 0, SD = 1
- C) Mean = 100, SD = 15
- D) Mean = 0.5, SD = 0.5
Show answer & explanation
Answer: B) Mean = 0, SD = 1
The Z-distribution is a special case of the normal distribution where the data has been transformed (standardized) so that the mean is exactly 0 and the standard deviation is exactly 1.
Question 52
Which probability distribution is strictly utilized for discrete events where there are only two possible outcomes per trial (e.g., success or failure)?
- A) Normal Distribution
- B) Binomial Distribution
- C) Poisson Distribution
- D) Continuous Uniform Distribution
Show answer & explanation
Answer: B) Binomial Distribution
The Binomial distribution is used for discrete experiments consisting of n independent trials, each having exactly two outcomes (success and failure), with a constant probability of success.
Question 53
What is the statistical formula used to convert a raw data point (X) into a standardized Z-score?
- A) Z = (X - μ) / σ
- B) Z = (μ - X) / σ
- C) Z = X × σ
- D) Z = (X + μ) / 2
Show answer & explanation
Answer: A) Z = (X - μ) / σ
The Z-score represents how many standard deviations a value is from the mean. It is calculated by subtracting the mean from the value and dividing the result by the standard deviation.
Question 54
In a Poisson Distribution, what specific condition must be true regarding the mean (μ) and the variance (σ²)?
- A) Mean > Variance
- B) Mean < Variance
- C) Mean = Variance
- D) Variance = Square Root of Mean
Show answer & explanation
Answer: C) Mean = Variance
A unique property of the Poisson distribution (often used for modeling the frequency of rare events over time) is that its mathematical mean and variance are perfectly equal.
Question 55
What is the total area under the entire curve of any continuous probability density function (like the Normal Distribution)?
- A) 0
- B) 0.5
- C) 1
- D) 100
Show answer & explanation
Answer: C) 1
Because the curve represents the total set of all possible outcomes, the total area integrated under the probability density function must mathematically equal 1.
Question 56
If an event has a probability of exactly 1.0, it is formally classified as a/an:
- A) Impossible Event
- B) Rare Event
- C) Likely Event
- D) Certain Event
Show answer & explanation
Answer: D) Certain Event
An event with a probability of 1.0 (or 100%) is mathematically guaranteed to occur.
Question 57
Which of the following is an example of an 'Objective' (Classical) probability assessment?
- A) A doctor estimating a 20% chance of a patient recovering based on experience.
- B) A financial analyst predicting a 60% chance of a stock market crash.
- C) Calculating a 1/6 (16.6%) chance of rolling a 4 on a fair, six-sided die.
- D) A weather forecaster predicting rain.
Show answer & explanation
Answer: C) Calculating a 1/6 (16.6%) chance of rolling a 4 on a fair, six-sided die.
Objective probability is based on logic or physical evidence (like the number of faces on a die) where the outcomes are clearly defined and equally likely.
Question 58
According to the properties of a Normal Distribution, approximately what percentage of data falls within ±2 standard deviations of the mean?
- A) 68%
- B) 90%
- C) 95.4%
- D) 99.7%
Show answer & explanation
Answer: C) 95.4%
The Empirical Rule states that roughly 95.4% of the area under the normal curve lies within two standard deviations of the arithmetic mean.
Question 59
In a Venn Diagram, the colored region representing everything inside circles A OR B is known as the:
- A) Intersection
- B) Union
- C) Complement
- D) Variance
Show answer & explanation
Answer: B) Union
The union of two sets (A ∪ B) represents all elements that reside in A, or in B, or in both simultaneously.
Question 60
When calculating the Expected Value E(X) of a discrete probability distribution, you must:
- A) Add all X values then divide by N.
- B) Multiply each outcome (X) by its probability [P(X)] and sum the results.
- C) Take the square root of the variance.
- D) Find the middle value in the sequence.
Show answer & explanation
Answer: B) Multiply each outcome (X) by its probability [P(X)] and sum the results.
The expected value is the long-term weighted average of a probability distribution, calculated as ∑ [X * P(X)].