Probability Questions and Solved Examples

Posted on May 2, 2025 in Statistics

Here is the giant combined cheat sheet with all questions, tables, charts, calculations, and answers included. You can copy this into a document editor and save it as a PDF.

Probability Questions and Answers

Question 1

Statement: Tossing a coin and getting a head, and then rolling a 6-sided die and getting a 5. Are these events dependent?

Answer: False

Explanation: The events are independent because the outcome of the coin toss does not affect the die roll.

Question 2

Which pairs of events are disjoint?

Answer:

• E1 = [Patient is dead], E2 = [Patient is alive]
• E1 = [A litter contains 10 mice], E2 = [A litter contains 15 or more mice]

Explanation: Disjoint events cannot occur simultaneously.

Question 3

Statement: 30% of flies are black, 70% are grey. 2 flies are chosen at random. What is the probability both are grey?

Answer: 0.49

Calculation: \( 0.7 \times 0.7 = 0.49 \)

Question 4

Statement: 30% of flies are black, 70% are grey. 2 flies are chosen at random. What is the probability of 1 grey and 1 black?

Answer: 0.42

Calculation: \( 2 \times (0.7 \times 0.3) = 0.42 \)

Question 5

Statement: A disease is inherited via sex-linked mode. Males have a 50% chance of inheriting it, females 0%. 51.3% of births are male. What is the probability a random child is affected?

Answer: 0.2565

Calculation: \( 0.513 \times 0.5 = 0.2565 \)

Question 6

Statement: Disease probability = 0.2. Test is positive 90% of the time for those with the disease and negative 75% of the time for those without. What is Pr{Test positive}?

Answer: 0.38

Calculation: \( (0.2 \times 0.9) + (0.8 \times 0.25) = 0.38 \)

Question 7

Statement: Compute Pr{Test positive | No disease}.

Answer: 0.25

Explanation: Test is positive 25% of the time for those without the disease.

Question 8

Statement: For a random person with a positive test, what is the probability they truly have the disease?

Answer: 0.4737

Calculation: \( \frac{0.2 \times 0.9}{0.38} = 0.4737 \)

Question 9

Statement: Compute Pr{ Y > 0 } for a continuous variable Y.

Answer: 0.5

Calculation: \( 0.3 + 0.15 + 0.05 = 0.5 \)

Question 10

Statement: Compute Pr{ Y = 0 } for a continuous variable Y.

Answer: 0

Explanation: For continuous variables, the probability at a single point is 0.

Question 11

Statement: Compute Pr{ |Y| > 3 } for a continuous variable Y.

Answer: 0.1

Calculation: \( 0.05 + 0.05 = 0.1 \)

3.2.1 Freshwater Sculpin Tail Vertebrae

Table:

Questions:

• a. What is the probability that the number of tail vertebrae equals 21?

  Answer: 51% (from the table).

• b. What is the probability that the number of tail vertebrae is less than or equal to 22?

  Answer: 3% + 51% + 40% = 94%.

• c. What is the probability that the number of tail vertebrae is greater than 21?

  Answer: 40% + 6% = 46%.

• d. What is the probability that the number of tail vertebrae is no more than 21?

  Answer: 3% + 51% = 54%.

3.3.1 Health Risk and Income

Table:

Questions:

• a. What is the probability that someone in this study smokes?

  Answer: 1,213 / 6,549 ≈ 0.185 (18.5%).

• b. What is the conditional probability that someone in this study smokes, given that the person has high income?

  Answer: 247 / 2,115 ≈ 0.117 (11.7%).

• c. Is being a smoker independent of having a high income? Why or why not?

  Answer: No, because Pr(Smokes) = 0.185 ≠ Pr(Smokes | High Income) = 0.117.

3.4.3 Trypanosoma Lengths

Density Curve Areas:

• Pr(Y < -3) = 0.05
• Pr(-3 < Y < -1) = 0.15
• Pr(-1 < Y < 0) = 0.3
• Pr(0 < Y < 1) = 0.3
• Pr(1 < Y < 3) = 0.15
• Pr(Y > 3) = 0.05

Questions:

• a. What is Pr(20 < length < 30)?

  Answer: 0.41 (from the density curve).

• b. What is Pr(length > 20)?

  Answer: 0.41 + 0.21 + 0.03 = 0.65.

• c. What is Pr(length < 20)?

  Answer: 0.01 + 0.34 = 0.35.

3.5.4 Fruitfly Body Color

Table:

Questions:

• a. What is Pr(Y ≥ 2)?

  Answer: 0.189 + 0.027 = 0.216.

• b. What is Pr(Y ≤ 2)?

  Answer: 0.343 + 0.441 + 0.189 = 0.973.

3.S.2 Centipedes in Beech Woods

Table:

Questions:

• a. What is Pr(Y = 1)?

  Answer: 36%.

• b. What is Pr(Y ≥ 2)?

  Answer: 14% + 4% + 1% = 19%.

3.2.3 College Students

Question:

In a college, 55% of students are women. If 2 students are sampled:

• a. What is the probability that both chosen students are women?

  Answer: 0.55 × 0.55 = 0.3025.

• b. What is the probability that at least 1 of the 2 students is a woman?

  Answer: 1 – (0.45 × 0.45) = 0.7975.

3.2.8 Medical Test

Question:

A test has 92% sensitivity and 94% specificity. Disease prevalence is 10%.

• a. What is the probability that a randomly chosen person will test positive?

  Answer: (0.1 × 0.92) + (0.9 × 0.06) = 0.146.

• b. What is the probability that a person who tests positive truly has the disease?

  Answer: (0.1 × 0.92) / 0.146 ≈ 0.6301.

3.3.2 Income and Smoking

Table: Same as 3.3.1.

Questions:

• a. What is the probability that someone is from the low-income group and smokes?

  Answer: 634 / 6,549 ≈ 0.0968.

• b. What is the probability that someone is not from the low-income group?

  Answer: (1,954 + 2,115) / 6,549 ≈ 0.621.

• c. What is the probability that someone is from the medium-income group?

  Answer: 1,954 / 6,549 ≈ 0.298.

• d. What is the probability that someone is from the low-income group or the medium-income group?

  Answer: (2,480 + 1,954) / 6,549 ≈ 0.677.

3.4.5 Trypanosoma Lengths (Sample of 2)

Questions:

• a. What is the probability that both trypanosomes will be shorter than 20 μm?

  Answer: 0.35 × 0.35 = 0.1225.

• b. What is the probability that the first trypanosome will be shorter than 20 μm and the second will be longer than 25 μm?

  Answer: 0.35 × 0.24 = 0.084.

• c. What is the probability that exactly 1 trypanosome will be shorter than 20 μm and 1 will be longer than 25 μm?

  Answer: 2 × (0.35 × 0.24) = 0.168.

3.5.5 Mean of Y (Fruitflies)

Question:

What is the mean number of black flies in a sample of 3?

Answer: μY = 0 × 0.343 + 1 × 0.441 + 2 × 0.189 + 3 × 0.027 = 0.9.

3.5.6 Standard Deviation of Y (Fruitflies)

Question:

What is the standard deviation of the number of black flies in a sample of 3?

Answer: σY = √[(0² × 0.343) + (1² × 0.441) + (2² × 0.189) + (3² × 0.027) – 0.9²] ≈ 0.87.

3.6.5 Snail Shell Color

Question:

In a population, 60% of snails have streaked shells. If 10 snails are sampled:

• a. What is the probability that 50% of the sample will have streaked shells?

  Answer: C(10,5) × (0.6)^5 × (0.4)^5 ≈ 0.2007.

• b. What is the probability that 60% of the sample will have streaked shells?

  Answer: C(10,6) × (0.6)^6 × (0.4)^4 ≈ 0.2508.

• c. What is the probability that 70% of the sample will have streaked shells?

  Answer: C(10,7) × (0.6)^7 × (0.4)^3 ≈ 0.2150.

3.6.6 Snail Shell Color (Sample of 10)

Questions:

• a. What is the mean number of streaked-shelled snails in a sample of 10?

  Answer: 10 × 0.6 = 6.

• b. What is the standard deviation of the number of streaked-shelled snails in a sample of 10?

  Answer: √(10 × 0.6 × 0.4) ≈ 1.55.

3.6.8 Infant Sex Ratio

Question:

The sex ratio of newborns is 105 males : 100 females. If 4 infants are sampled:

• a. What is the probability that 2 are male and 2 are female?

  Answer: C(4,2) × (105/205)^2 × (100/205)^2 ≈ 0.374.

• b. What is the probability that all 4 are male?

  Answer: (105/205)^4 ≈ 0.067.

• c. What is the probability that all 4 are the same sex?

  Answer: (105/205)^4 + (100/205)^4 ≈ 0.121.

3.S.4 Centipedes (Sample of 5 Squares)

Question:

What is the expected number of squares that contain at least 1 centipede?

Answer: 5 × (1 – 0.45) = 2.75.

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