Statistics Practice: Probability and Random Variables

Compute the union, complement, and difference probabilities for Visa and Mastercard cards: P(A∪B)=0.65, P(neither)=0.35, and P(A∩B^c)=0.25, illustrated with a venn diagram.

Solve a sequence of probability problems involving unions, complements, and De Morgan's law, using Venn diagrams to compute intricate event probabilities.

Apply probability principles to three options: compute P(A∪B∪C)=0.85, P(none)=0.15, and derive the probability of selecting exactly one option using complements, De Morgan's laws, and a Venn diagram.

this lecture solves five dryer purchases, deriving the probability of at least two electric purchases from at most one electric 0.428, and 0.879 for at least one of each type.

Compute the probability of no type 1 defect (0.8), and the probability of both type 1 and type 2 defects (0.06) using inclusion–exclusion, and at most two defects (0.99).

Work through a solved example with mutually exclusive and exhaustive events A, B, and C; use union probability equals one and inclusion-exclusion to determine P(A) = 4/13.

Five urns (phones) each contain ten balls with increasing defect counts. Using total probability, the probability of drawing a defective ball is 3/10, with a follow-up on P(U2|D).

Apply Bayes' rule to compute P(B|A) with 0.001 prevalence, 0.999 test positive given TB, and 0.002 test positive given no TB; the result is 0.333.

Apply Bayes' theorem to a two-urn problem with red and blue balls, and equal urn choice, computing the probability the red ball came from urn two as two thirds.

Apply Bayes' rule to determine probability the machine is correctly set up given an acceptable item. With a prior of 0.80 and conditional probabilities 0.90 and 0.30, P(C|A) = 0.923.

Determine the probability that the sum of two balanced dice is even by using independence and the even/odd cases, yielding 1/2.

Examine independence of coin-toss events by computing intersections and individual probabilities, showing C and B are independent while A and G are not, and illustrating the independence criterion P(X∩Y)=P(X)P(Y).

Solve a two-event probability problem with independent events A and B given P(A∩B)=1/8 and P(A^c∩B^c)=3/8, deducing P(A)=1/4 and P(B)=1/2.

Compute the probability that three independent students fail to solve a problem using their solve probabilities 3/7, 3/8, and 1/3, yielding 5/21.

Learn two methods to find the probability of at least one six when rolling two dice: count favorable outcomes (11/36) or use 1 minus no-six probability.

Solve a probability problem with three circuits, where X is independent of Y and Z and Y and Z interdependent, to find the probability that only Y works (0.02).

Solve a probability puzzle about two letters with a ladies first policy, and compute the probability that the second letter is from a female given the first is female.

Analyze pairwise independence of E, F, and G, apply conditional probability to compute P(E^c ∩ F^c | G) using the given E ∩ F ∩ G = 0.

solves a conditional probability problem with two coins, where x is the heads from three tosses and y is the heads from coin tossed x times, computing p(x≥2 | y=1).

Four testers independently rank three brands A, B, and C at random. We compute the probability that chocolate E’s total score equals four by summing the four independent ranks.

Learn to compute the probability mass function for the days until both magazines arrive using independent daily probabilities, then apply PMF concepts to airline overbooking and standby scenarios.

Solve two discrete-probability problems using the given pmf for lines in use, and transform to not in use with 6 minus x, then verify the pmf sums to one.

Examine quality-control sampling with two from five, derive the probability distribution for defective items, and compute the cumulative distribution function; compare two real-world problems about inspections and insurance.

solve a two-battery test with 90% acceptance, derive P(V_i = i−1), and evaluate i = 2, 3, 5; then find the pmf and cdf of M, maximum of two dice.

Part a derives the pmf of x, the number of students with their own book. Part b computes the pmf, cdf, and P(2 ≤ x ≤ 6) for late arrivals.

Analyze ben folds law as a legitimate pmf for leading digits 1–9 and compare it with a discrete uniform distribution. Determine the cumulative distribution function and relevant probabilities.

Solve a probability problem for a random variable X representing car defects by applying the cumulative distribution function to compute P(X=2), P(X>3), and P(2≤X≤5).

Derive the probability mass function from the cumulative distribution function, compute point probabilities, and apply cdf differences to find ranges like P(3≤X≤6) and P(X≥4).

Compute the expected storage space X, its square, and variance for three freezer models; apply linear transformations to price and assess price variance, then evaluate the actual capacity.

Compute E[X^2] and derive Var(X) for a discrete pmf, and verify E[X] and E[X^79] for the first problem. Then establish E[X] is finite when B(X)=c/x^2 for x≥1.

Compare the expected net revenue of ordering three versus four copies by applying the demand probability mass function, costs, and price, and conclude to order four.

Compute the insurer’s premium for a $500 deductible by analyzing payouts at x = 0, 5,000, and 10,000 with given probabilities to yield a $100 expected profit, a $700 premium.

Compute the expected value and variance of X from its pmf for a fair die, then assess the expected pounds left and its variance after 5-pound batch orders.

Compute e[x^2] from e[x(x-1)]=27.5 and e[x]=5, yielding e[x^2]=32.5 and var(x)=7.5. Relate e[x], e[x(x-1)], and var(x) and apply a ≤ x ≤ b to conclude a ≤ e[x] ≤ b.

Calculate the player's expected gain by evaluating all two-draw outcomes from three $1 chips and two $4 chips without replacement, yielding E[X] = 4.4.

The variance of x equals 2b, since the expected value is 1 and the expected value of x squared is 1+2b, so maximum variance occurs at b=0.5.

solve a binomial problem with p=0.3 and n=25 to get the mean 7.5 and std dev 2.29. compute the probability beyond two standard deviations, assess stock constraints, and estimate revenue.

Solve two binomial practice problems: exactly one second among six goblets and the expected toll revenue for 25 vehicles with 60% passenger cars.

Use binomial models to compare topic a and b for interlibrary loan book arrivals under p=0.9 and p=0.5, and compute mean and variance for debit card adoption among 100 customers.

Derive binomial distribution parameters from given probabilities for n = 5 by applying the binomial probability expression to X = 1 and X = 2, yielding p = 0.2.

Explore a binomial model with independent Bernoulli trials and X ~ binomial(n, B); by equating P(X = k) and P(X = k+1), identify the true statement, concluding option D.

This solved example verifies a probability density function integrates to one, then uses a transformed pdf to compute probabilities such as p(x<=200) and p(100<=x<=200) with t=100.

Determine the normalizing constant k for the probability density function of X on [0,2], where k = 3/8, and compute P(X≤1)=0.125, P(1≤X≤1.5)=0.297, P(X≥1.5)=0.578.

Analyze a bus waiting-time problem with a piecewise pdf derived from uniform distribution, verify the pdf integrates to one, and compute probabilities for total waiting time in various intervals.

Graph the density on [3,5] and verify its area equals one. Compute probabilities by integration, including x ≤ 4, 3.5 ≤ x ≤ 4.5, and x > 4.5.

Determine the normalization constant a for the pdf of a continuous variable x with x > 0 by enforcing ∫_0^∞ a e^{-x/5} dx = 1, yielding a = 1/5 (0.2).

The lecture solves finding the median lambda for a continuous variable with pdf f(x)=3x^2 on [0,1], by setting F(λ)=0.5 and evaluating ∫0^λ 3x^2 dx = 0.5, yielding λ=(1/2)^(1/3).

Compute p(a∪b) for x with density f(x)=1/x^2 on [1,2] and [4,5], noting a∩b=∅ and obtaining p(a∪b)=1/2+1/20=11/20.