Probability Basics
A coach says, "I think we have about a 60% chance of winning tonight." Which approach to probability is the coach using: theoretical, empirical, or subjective?
Subjective. It is based on personal judgment and experience, not on equally likely outcomes or on observed data.
Can a single playing card be both a heart and a club? What do we call two events that cannot happen at the same time?
No. Events that cannot occur together are mutually exclusive: P(A AND B) = 0.
In medical testing, what is a false positive?
A positive test result for someone who does not actually have the condition.
Is each random variable discrete or continuous? (a) the number of texts you send today (b) the time you spend texting today.
(a) Discrete, because it is counted. (b) Continuous, because it is measured.
X is a continuous random variable. What is P(X = 7), and why?
0. For a continuous variable, probability is area under the curve, and a single point has no width.
The probability that a randomly chosen SVU student owns a car is 0.38. What is the probability that the student does not own a car?
0.62, by the complement rule: P(A′) = 1 − P(A) = 1 − 0.38.
A and B are mutually exclusive, with P(A) = 0.25 and P(B) = 0.40. Find P(A OR B).
0.65. For mutually exclusive events, P(A OR B) = P(A) + P(B).
A screening test correctly flags 45 of the 50 people who have a disease. What is the test's sensitivity, and what does it measure?
45/50 = 0.90. Sensitivity is P(test positive | has the disease): how well the test catches the people who have it.
A probability distribution has x = 0, 1, 2, 3 with P(x) = 0.20, 0.35, 0.30, and k. What must k be for this to be a valid distribution?
k = 0.15. The probabilities must add to 1: 1 − (0.20 + 0.35 + 0.30) = 0.15.
Song lengths on a playlist are uniformly distributed between 2 and 6 minutes. Find P(a song is longer than 5 minutes) and the mean song length.
X ~ U(2, 6). P(X > 5) = (6 − 5)/(6 − 2) = 0.25. Mean = (2 + 6)/2 = 4 minutes.
A softball player got 70 hits in 250 at-bats this season. What is the empirical probability that she gets a hit in her next at-bat, and why is it called empirical?
70/250 = 0.28. It is empirical because it comes from observed data (relative frequency), not from equally likely outcomes.
P(A) = 0.50, P(B) = 0.35, and P(A AND B) = 0.15. Find P(A OR B).
Addition Rule: 0.50 + 0.35 − 0.15 = 0.70.
Of 120 runners, 70 lift weights daily: 50 of them set a personal record (PR) this season and 20 did not. Of the 50 who do not lift, 20 set a PR and 30 did not. Find P(PR) and P(lifts AND PR).
P(PR) = (50 + 20)/120 = 70/120 ≈ 0.583. P(lifts AND PR) = 50/120 ≈ 0.417.
A raffle ticket costs $5. The only prize is $100, and each ticket has a 0.02 chance of winning it. What is the expected net gain per ticket?
E(net) = 100(0.02) − 5 = −$3. On average a buyer loses $3 per ticket.
The wait for a campus shuttle is exponential with a mean of 4 minutes. Find P(you wait more than 6 minutes).
m = 1/4 = 0.25, so P(X > 6) = e^(−0.25·6) = e^(−1.5) ≈ 0.223.
P(A AND B) = 0.21 and P(A) = 0.35. Find P(B | A).
P(B | A) = P(A AND B)/P(A) = 0.21/0.35 = 0.60.
P(A) = 0.6, P(B) = 0.5, and P(A AND B) = 0.3. Are A and B independent? Are they mutually exclusive?
Independent, because P(A)·P(B) = 0.6 × 0.5 = 0.30 = P(A AND B). Not mutually exclusive, because P(A AND B) is not 0.
Using the runners in the 300 clue, find P(PR | lifts daily). Are lifting and setting a PR independent?
P(PR | lifts) = 50/70 ≈ 0.714. That is not equal to P(PR) ≈ 0.583, so they are not independent.
20% of a café's customers order dessert, independently. For the next 10 customers, find P(exactly 3 order dessert), and give the mean and standard deviation of the number who do.
X ~ B(10, 0.2). P(X = 3) = BINOM.DIST(3, 10, 0.2, FALSE) ≈ 0.201. μ = np = 2; σ = √(np(1−p)) = √1.6 ≈ 1.26.
Exam scores are N(70, 8). A student scores 86. Find the z-score and P(X > 86). Is 86 unusually high?
z = (86 − 70)/8 = 2. P(X > 86) = 1 − NORM.DIST(86, 70, 8, TRUE) ≈ 0.023. Yes, it is about two standard deviations above the mean, and only about 2% of scores are higher.
A house has three smoke detectors that work independently, and each one works in a fire with probability 0.90. What is the probability that at least one detector works?
Use the complement: P(none work) = 0.10³ = 0.001, so P(at least one works) = 1 − 0.001 = 0.999.
A bag holds 5 red and 7 blue marbles. Two are drawn without replacement. Find the probability that both are blue.
General Multiplication Rule: (7/12)(6/11) = 42/132 ≈ 0.318. The second draw has one fewer blue marble and one fewer marble in total.
A disease affects 5% of a population. A test has 90% sensitivity and 94% specificity. Use a hypothetical group of 10,000 people to find the positive predictive value (PPV).
500 have the disease, giving 450 true positives. 9,500 do not, and 6% of them (570) test positive falsely. PPV = 450/(450 + 570) = 450/1,020 ≈ 0.44: fewer than half of the positive results are real.
A copy editor finds an average of 3 typos per page, and typos occur independently. Find P(at most 1 typo on a page) and P(more than 4 typos on a page).
X ~ P(3). P(X ≤ 1) = POISSON.DIST(1, 3, TRUE) ≈ 0.199. P(X > 4) = 1 − POISSON.DIST(4, 3, TRUE) ≈ 1 − 0.815 = 0.185.
Battery life is N(500, 40) hours. The maker will replace any battery whose life is in the bottom 5%. What is the cutoff, and what Excel formula finds it?
=NORM.INV(0.05, 500, 40) ≈ 434.2 hours. Batteries lasting less than about 434 hours are replaced.