FE Other Disciplines (NCEES Fundamentals of Engineering) — All Questions

Two fair six-sided dice are rolled. What is the probability that the sum equals 7?

• a.1/12
• b.1/9
• c.1/8
• d.1/6✓

There are 36 equally likely outcomes. The combinations summing to 7 are (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6 outcomes. Probability = 6/36 = 1/6. Seven is the most likely sum for two dice.

Find the sample standard deviation of the data set 2, 4, 6, 8, 10.

• a.3.16✓
• b.6.32
• c.2.83
• d.10

The mean is 6. Deviations are -4, -2, 0, 2, 4; squared they are 16, 4, 0, 4, 16 summing to 40. Sample variance divides by (n - 1) = 4, giving 10; the standard deviation is sqrt(10) = 3.16. Dividing by n-1 (not n) gives the unbiased SAMPLE estimate.

A fair six-sided die is rolled once. What is the probability of rolling a number greater than 4?

• a.1/3✓
• b.1/6
• c.1/2
• d.2/3

The outcomes greater than 4 are 5 and 6, i.e. 2 of 6 equally likely outcomes. Probability = 2/6 = 1/3. Count favorable outcomes over total outcomes.

Two fair six-sided dice are rolled. What is the probability that the sum equals 8?

• a.6/36
• b.5/36✓
• c.4/36
• d.1/6

The combinations summing to 8 are (2,6),(3,5),(4,4),(5,3),(6,2) = 5 outcomes out of 36. Probability = 5/36. There are fewer ways to make 8 than to make 7.

A single card is drawn from a standard 52-card deck. What is the probability it is a king?

• a.1/13✓
• b.4/13
• c.1/52
• d.1/4

There are 4 kings among 52 cards, so the probability is 4/52 = 1/13. Reducing the fraction 4/52 by dividing by 4 gives 1/13.

A fair coin is flipped 3 times. What is the probability of getting exactly 2 heads?

• a.1/8
• b.1/2
• c.1/4
• d.3/8✓

Using the binomial formula, P = C(3,2)·(0.5)^2·(0.5)^1 = 3·(1/8) = 3/8. There are 3 arrangements (HHT, HTH, THH) out of 8 equally likely outcomes.

A binomial experiment has n = 20 trials with success probability p = 0.3. What is the mean number of successes?

• a.0.3
• b.4.2
• c.6✓
• d.14

The mean of a binomial distribution is n·p = 20·0.3 = 6. The value 4.2 is the variance n·p·(1-p), not the mean.

A binomial experiment has n = 100 trials with success probability p = 0.5. What is the variance of the number of successes?

• a.5
• b.50
• c.100
• d.25✓

The binomial variance is n·p·(1-p) = 100·0.5·0.5 = 25. The value 5 is the standard deviation (its square root), and 50 is the mean n·p.

In how many distinct orders can 4 different books be arranged on a shelf?

• a.16
• b.256
• c.24✓
• d.12

The number of arrangements (permutations) of 4 distinct items is 4! = 4·3·2·1 = 24. Each position reduces the remaining choices by one.

How many ways can a committee of 2 be chosen from 6 people?

• a.30
• b.12
• c.15✓
• d.36

Order does not matter, so use combinations: C(6,2) = 6!/(2!·4!) = (6·5)/2 = 15. Dividing 30 (the permutation count) by 2! removes duplicate orderings.

Given P(A and B) = 0.2 and P(B) = 0.5, what is the conditional probability P(A given B)?

• a.0.1
• b.0.4✓
• c.0.7
• d.0.25

The definition of conditional probability is P(A|B) = P(A and B)/P(B) = 0.2/0.5 = 0.4. It rescales the joint probability by the probability of the conditioning event.

Events A and B are independent with P(A) = 0.3 and P(B) = 0.4. What is P(A and B)?

• a.0.1
• b.0.7
• c.0.12✓
• d.0.35

For independent events P(A and B) = P(A)·P(B) = 0.3·0.4 = 0.12. Multiplication (not addition) gives the joint probability of independent events; 0.7 would be their sum.

What is the arithmetic mean of the data set 3, 7, 7, 19, 4?

• a.7
• b.8✓
• c.10
• d.40

The mean is the sum divided by the count: (3 + 7 + 7 + 19 + 4)/5 = 40/5 = 8. The mean uses every value, unlike the median or mode.

What is the median of the data set 3, 7, 2, 9, 5, 4?

• a.4
• b.5
• c.3.5
• d.4.5✓

Sorting gives 2, 3, 4, 5, 7, 9. With an even count of six values, the median is the average of the two middle values: (4 + 5)/2 = 4.5.

What is the mode of the data set 2, 4, 4, 4, 5, 6, 7?

• a.7
• b.4✓
• c.4.57
• d.5

The mode is the most frequently occurring value. Here 4 appears three times, more than any other value, so the mode is 4. (The mean, about 4.57, differs from the mode.)

For a normal distribution, approximately what percentage of data falls within one standard deviation of the mean?

• a.99.7%
• b.95%
• c.50%
• d.68%✓

By the empirical (68-95-99.7) rule, about 68% of values lie within +/-1 standard deviation, 95% within +/-2, and 99.7% within +/-3 of the mean.

For a normal distribution, approximately what percentage of data falls within two standard deviations of the mean?

• a.68%
• b.90%
• c.99.7%
• d.95%✓

By the empirical rule, about 95% of values lie within +/-2 standard deviations of the mean. About 68% lie within +/-1 and 99.7% within +/-3.

A value of 85 comes from a normal distribution with mean 70 and standard deviation 10. What is its z-score?

• a.1.5✓
• b.-1.5
• c.15
• d.0.67

The z-score is z = (x - mean)/sigma = (85 - 70)/10 = 15/10 = 1.5. It expresses how many standard deviations a value lies above the mean.

A game pays $10 with probability 0.2 and costs (loses) $2 with probability 0.8. What is the expected value per play?

• a.$2.00
• b.-$1.60
• c.$0.40✓
• d.$1.60

Expected value = sum of (outcome x probability) = (10)(0.2) + (-2)(0.8) = 2 - 1.6 = $0.40. A positive expected value means the game favors the player on average.

A Poisson process has a mean of 3 events per interval. What is the probability of observing zero events in an interval?

• a.0.15
• b.0.0498✓
• c.0
• d.0.3

The Poisson probability is P(X=k) = (lambda^k · e^(-lambda))/k!. For k = 0: P = e^(-3) = 0.0498. Any quantity raised to the 0 power is 1 and 0! = 1.

For a Poisson distribution with a mean of 4, what is the variance?

• a.2
• b.8
• c.4✓
• d.16

A defining property of the Poisson distribution is that its variance equals its mean: variance = lambda = 4. The standard deviation would be sqrt(4) = 2.

What is the sample variance of the data set 2, 4, 6?

• a.4✓
• b.2.67
• c.8
• d.2

The mean is 4; deviations are -2, 0, 2 with squares 4, 0, 4 summing to 8. Sample variance divides by (n - 1) = 2, giving 8/2 = 4. Dividing by n instead would give the biased 2.67.

A population has standard deviation 20. For a sample of size 100, what is the standard error of the mean?

• a.2✓
• b.20
• c.0.2
• d.0.02

The standard error is sigma/sqrt(n) = 20/sqrt(100) = 20/10 = 2. Larger samples reduce the standard error and sharpen the estimate of the mean.

How many ordered arrangements (permutations) of 2 items can be selected from 5 distinct items?

• a.120
• b.10
• c.25
• d.20✓

Permutations P(5,2) = 5!/(5-2)! = 5·4 = 20. Order matters here, so this is twice the combination count C(5,2) = 10.

A fair coin is flipped twice. What is the probability of getting at least one head?

• a.0.5
• b.1.0
• c.0.25
• d.0.75✓

Use the complement: P(at least one head) = 1 - P(no heads) = 1 - (0.5)(0.5) = 1 - 0.25 = 0.75. The complement rule is easier than summing the favorable cases.

Events A and B are mutually exclusive with P(A) = 0.5 and P(B) = 0.3. What is P(A or B)?

• a.0.65
• b.0.8✓
• c.0.2
• d.0.15

For mutually exclusive events P(A or B) = P(A) + P(B) = 0.5 + 0.3 = 0.8, because they cannot occur together and there is no overlap to subtract.

For events with P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.2, what is P(A or B)?

• a.1.1
• b.0.9
• c.0.3
• d.0.7✓

The general addition rule is P(A or B) = P(A) + P(B) - P(A and B) = 0.5 + 0.4 - 0.2 = 0.7. Subtracting the overlap avoids double-counting.

If the probability that an event occurs is 0.35, what is the probability that it does not occur?

• a.0.65✓
• b.1.35
• c.0.5
• d.0.35

By the complement rule P(not A) = 1 - P(A) = 1 - 0.35 = 0.65. The probabilities of an event and its complement always sum to 1.

A student scores 80 on an exam weighted 30% and 90 on an exam weighted 70%. What is the weighted mean score?

• a.86
• b.87✓
• c.170
• d.85

The weighted mean is sum of (value x weight) = 80(0.3) + 90(0.7) = 24 + 63 = 87. Because the higher score carries more weight, the result exceeds the simple average of 85.

A continuous uniform distribution spans the interval [2, 8]. What is its mean?

• a.10
• b.5✓
• c.6
• d.3

The mean of a continuous uniform distribution is (a + b)/2 = (2 + 8)/2 = 5. By symmetry the mean sits at the midpoint of the interval.

A continuous uniform distribution spans the interval [0, 12]. What is its variance?

• a.12✓
• b.6
• c.24
• d.144

The variance of a continuous uniform distribution is (b - a)^2/12 = (12)^2/12 = 144/12 = 12. The divisor 12 is specific to the uniform distribution.

Which correlation coefficient indicates the strongest linear relationship between two variables?

• a.0.10
• b.0.85
• c.0.50
• d.-0.92✓

Correlation strength depends on the magnitude (absolute value), not the sign. |−0.92| = 0.92 is closest to 1, so -0.92 indicates the strongest (negative) linear relationship.