Complete Math, Statistics & Probability for Machine Learning

Explore how combinatorics fuels machine learning, from neural architecture search and auto ml to explainable AI and Bayesian optimization.

Explore factorial, the product of integers from 1 to n, and see how it counts arrangements of distinct objects, with examples like 3! = 6 and 0! = 1.

Master factorial concepts through five exercises and solutions, including simplifying factorial expressions, comparing results, and solving n! problems to reinforce permutation techniques.

Demystify the difference between permutation and combination by contrasting arrangement with selection, and apply factorial-based formulas to compute permutation and combination from n and r.

Clarifying permutation versus combination, this lecture emphasizes when order matters and applies factorial-based formulas to examples like selecting top finishers, forming passwords, ice cream orders, and teams.

Explore cyclic permutations by arranging objects around a circle using one fixed object, yielding (n-1)!, illustrated with five people around a circular table: 4! = 24.

Explore cyclic permutation for seating six friends around a circular table, where the number of arrangements equals (n-1)!, giving 5! = 120.

Explore cyclic permutation by counting ways to arrange seven distinct books on a circular shelf and applying a factorial-based calculation.

Explore cyclic permutation with two chairs must be next to each other: merge the pair into one entity around a circle, giving 7! arrangements, and double for the internal order.

Explore digits permutation to form four-digit numbers from digits 1 to 9, with and without repetition. Apply position-by-position counting, including an even last digit when repetition is not allowed.

Form four-digit numbers using digits 1–4 without repetition, explaining that the total count is 4 factorial, which equals 24 permutations.

Form three-digit numbers from digits 1, 2, and 3 without repetition, demonstrating that the count equals six.

Compute the number of 4-digit numbers formed from digits 0, 1, 2, 3 without repetition and not starting with zero, resulting in 18 possibilities.

Form five-digit numbers using digits 1, 3, 5, 7, and 9 without repetition; every permutation yields an odd number, totaling 5! possible arrangements.

Form four-digit numbers from digits 1 to 4 with no repetition. Compute the count using four factorial, which equals 24.

Explore the permutation of the word mango and count all possible arrangements with no constraints. The result is 5! = 120.

Explore permutations of letters with the vowels a and o treated as a block, using factorial calculations and multiple grouping cases to illustrate probability ideas.

Demonstrate how to count distinct permutations of the word mathematics by accounting for repeated letters, using 11! divided by 2! for m, 2! for a, and 2! for t.

Explore letters permutation with repetitions by counting arrangements of the word cheese, applying factorials, and dividing by repeated letters, in exercise and solution 4.

Compute permutations of the word table and apply a vowel-at-beginning constraint, using factorials (2! and 3!) to determine 12 valid arrangements.

Explore permutations with repetition using the word rabbit. Count 360 total arrangements and subtract cases where the two b's are together to yield 240 valid orders.

Explore letter permutations of giraffe using factorials, first with no repetition, then with the two f's together as a single unit, giving 6! 720.

Explore row arrangement permutation by analyzing how four students can be seated in a row of four chairs. Use factorial to compute the total arrangements, which equals 24.

Explore row-arrangement permutation by solving how five different books can be arranged on a shelf, with a practical exercise and solution.

Learn how to count ordered selections of three officers from six candidates using permutation theory and factorials, yielding 6P3 = 120.

Explore row arrangement permutation by seating three boys and three girls in six chairs with alternating positions, revealing 72 possible arrangements.

Explore row arrangement permutations by seating eight people in eight chairs with the first chair fixed, yielding seven factorial arrangements.

Count how many ways musicians can sit in a row when the violinist and cellist must sit adjacent; multiply the parts to get 1,440 total arrangements.

Compute the number of arrangements for six people in a row when A and B cannot sit next to each other using the block method: 6! minus 2×5! equals 480.

Analyze seating permutations for a four-person committee with the chair fixed at seat one and the vice chair fixed at seat two, leaving two members to arrange, yielding two arrangements.

Use combinations to determine how many ways to select a committee of three from ten, noting this is a selection, not an arrangement; the calculation yields 120.

Explore how to calculate combinations by selecting four students from a class of 25 using the 25 choose 4 formula, yielding 12,650 possible selections.

Count the ways to draw three balls from a box of eight red and five blue with at least one blue, using 13 choose 3 minus 8 choose 3.

Explore combinatorial reasoning by calculating how many ways to choose five toppings from 12, using the 12 choose 5 combination and factorial simplifications.

Form a four-person committee from a group of six men and seven women with at least two women, by counting the 2w-2m, 3w-1m, and 4w-0m cases to total 560.

Apply the combination approach to count the number of ways to select five players from twelve with the captain included, yielding 330 possible starting lineups.

This exercise counts committees of four from eight men and four women with at least one man and one woman, using combinations: 3+1, 1+3, and 2+2, totaling 424.

Count four marbles from five red, four blue, and three green with at least one red. Break into cases (one to four red) and compute totals with combinations, yielding 460.

Explore forming a four-person committee from ten students (freshmen, sophomores, juniors) with at least one from each group. Compute distributions and combinations, arriving at 126 possible committees.

Learn to compute factorials in Python using the math and scipy libraries, including 5 factorial, compare integer results with decimal outputs, and understand how to control formatting.

Explore Python for permutation using the math and scipy libraries to compute nPr, factorial, and arrangements; practice with four-digit numbers without repetition and circular seating of six people.

Compute combinations in Python with math and SciPy, using comb and exact=true for integers, and apply to 12 choose 6, 1000-sample 70-30 train-test splits, and ensemble parameter selections.

Explore the role of probability in machine learning, highlighting its importance before delving into probability concepts.

Probability measures the likelihood of events under uncertainty, expressed as a number between zero and one. Illustrate fair coin tosses, a six-sided die, and drawing from a 52-card deck.

Explore basic probability terms such as experiment, outcome, sample space, omega, and event, and apply the probability formula to real examples like dice and coin tosses.

Compute the probability of drawing a non-red ball from a mix of eight red, four blue, and two green balls, yielding 3/7.

Compute the probability of maths passing and English failing as P × (1 − Q) = P − P Q, using the and rule (multiplication) and one minus Q.

Calculate how many red balls to add to make red probability become two over three, starting from six red out of fifteen. Set (6+x)/(15+x)=2/3 and solve to x=12.

Use the total probability axiom to solve for A. Compute P(A)=1−(1/4+1/5+1/6) with P(X)=1, yielding 23/60.

Compute the probability of rolling a five with a perfect die; since there are six possible outcomes, the probability is one out of six.

Calculate the probability of drawing a blue ball from a bag of nine balls with three blue, two white, and four red using direct counting and the complement rule.

Explore probability with a fair die to determine the likelihood of rolling an odd number, illustrating that three odd outcomes among six yield one half probability.

When today is Tuesday, the probability that tomorrow will be Wednesday is 1 (100 percent), illustrating a deterministic next-day event.

Compute the probability of drawing an ace from a standard deck; four aces exist in 52 cards, so the probability is 4/52, which simplifies to 1/13.

Calculate the probability of drawing a green marble from a jar with five red, three green, and two blue marbles. The probability is 3/10.

Calculate the probability of rolling a six on the first roll and an even number on the second roll, multiplying 1/6 by 3/6 to get 1/12.

Compute the probability that a randomly selected student is a girl in a class of 35, where 30 are boys and 12 are guests, guiding learners from question to solution.

Explore theoretical probability, calculated without experiments, using existing knowledge about the physical situation. See how juridical probability predicts event likelihood when empirical data is difficult or impossible to collect.

Explore a theoretical probability exercise on selecting two numbers from a set and determining when their product is even, using a 3x3 outcome table, yielding a probability of 8/9.

Compute the probability that a random number from 31 to 49 is a multiple of seven, counting 19 numbers and identifying three multiples (35, 42, 49) to get 3/19.

Toss three coins and compute the probability of at least two heads. Count the eight equally likely outcomes, identify four favorable cases, and conclude the probability is 1/2.

Compute the probability that the sum of two fair dice is at least nine by counting the 10 favorable outcomes among 36 total, giving 5/18.

Theoretical probability shows that tossing a coin and a die yield a tail and a number less than four with 12 total outcomes and 3 favorable ones, giving 1/4.

Calculate the probability of getting at least two tails when tossing three coins, using combinatorial counts and a worked exercise solution.

compute the probability of a sum of five when rolling two fair six-sided dice, showing four favorable outcomes out of 36, i.e., 4/36 equals 1/9.

Compute the probability that the first roll is four and the second is odd on a six-sided die. Show that 3 of 36 outcomes are favorable, giving 1/12.

Compute the probability that the first die shows an even number and the second die a multiple of three, using two rolls, yielding 8 out of 64, i.e., 1/8.

Explore empirical probability, also known as experimental probability or relative frequency, based on observed data and the ratio of event occurrences to total trials for cases where theory is difficult.

Explain empirical probability by flipping a coin 200 times and using 115 heads to compute an empirical probability of heads, 115/200, simplified to 23/40.

Compute the empirical probability that a randomly selected person prefers tea, given a survey of 500 respondents with 280 favoring tea, yielding 280/500 or 14/25.

Compute empirical probability from the experiment where a six appeared 30 times out of 150 trials. Derive the probability as 30 over 150.

Compute the empirical probability of a basketball player making a free throw from the given data, yielding 13/20 as the estimated probability.

Compute the empirical probability of drawing a red marble after 30 draws and compare it to the theoretical probability from 40 red out of 70 total, yielding 8/15 and 4/7.

Compute the empirical probability that a random student owns a bike. Show that with 80 of 200 students owning bikes, the probability equals 80/200, i.e., 2/5.

Learn how to compute empirical probability from a data set by analyzing 50 trials with 10 successes, yielding a probability of 1/5 for landing on the green section.

Compute the empirical probability of receiving a spam email from 300 emails, where 90 are spam, resulting in 3/10.

Calculate the empirical probability of a defective light bulb in a batch of 180 with 50 defective items.

Explore an exercise on empirical probability using a frequency table to find the chance a random person prefers a cat, with the solution showing total frequency and seven over twenty.

Explore an empirical probability exercise using a 50-game point distribution to compute the probability of winning 20 points.

Compute empirical probability from a grade frequency table to find the chance a randomly chosen student earns a B, showing 30 out of 100 equals 0.3.

Compute the empirical probability that an accident in a factory was caused by faulty equipment using the annual table, yielding 0.25 (one over four) as the result.

Compute the empirical probability of a billing issue from the weekly distribution of customer calls, yielding a total of 12/35.

Calculate empirical probability by using car color frequencies from a table; determine the probability of selecting a red car as 7/36, with a clear step-by-step solution.

Compute the empirical probability of choosing a mystery book by dividing its frequency (90) by the total book frequency and expressing the quotient as a decimal.

Calculate the empirical probability of spring as 30 out of 120 respondents, using the frequency over total to illustrate the concept.

Compute empirical probability from a frequency table by dividing the meat flavor count (20) by the total (80) to obtain 1/4, illustrating basic probability with a chocolate example.

Apply empirical probability to a family data table with 74 cases, computing probabilities for two children, three or more, and less than two.

Compute the empirical probability of obtaining a prime number from the given values in the table, by totaling 19, 24, 12, and 22 over 100, yielding 65/100 (13/20).

Calculate the empirical probability that a randomly selected team scores at most four goals using a frequency table of 32 World Cup teams; the result is 20/32 or five-eighths.

explore an empirical probability exercise by summing observed values to 21, computing the non-outcome probability as (5+7)/21 and simplifying to 4/7 using the complement rule.

Apply the addition rule to mutually exclusive events, where A or B cannot occur together, so the probability of A or B equals the sum of their individual probabilities.

Utilize the addition rule for mutually exclusive events to two dessert options, cake and ice cream; with cake probability 0.4, ice cream probability is 0.6.

Apply the addition rule for mutually exclusive events to find humanities probability as 0.25 given engineering 0.4 and business 0.35, using P(H) = 1 − P(E) − P(B).

Calculate the probability that the sum of two rolled dice is 7 or 11, using a 36-outcome table to count six outcomes for 7 and two for 11, yielding 2/9.

Apply the addition rule for mutually exclusive events to find the probability of selecting a student from group two or group three in a 35-student class, yielding 6/7.

Apply the addition rule for mutually exclusive events to calculate the probability of selecting a male student from group two or group three, using the gender table and group counts.

Apply the addition rule for mutually exclusive events to compute the probability of scoring 2 or 3, by summing 7/22 and 5/22 from the table with total frequency 22.

Explore the addition rule for mutually exclusive events through an exercise on numbers 15–29, showing the probability of a number being a multiple of seven or a prime, 7/18.

Apply the addition rule for non-mutually exclusive events to find P(A or B) as P(A) plus P(B) minus P(A ∩ B), noting the intersection is not empty.

Apply the addition rule for non-mutually exclusive events to find the probability of participating in either sport or arts. With P(S)=0.60, P(A)=0.50, P(S ∩ A)=0.20, compute P(S ∪ A)=0.90.

Apply the addition rule for non-mutually exclusive events. Using 120 coffee, 80 tea, and 40 both out of 200, the probability of coffee or tea is 160/200 (4/5).

Apply the addition rule for non-mutually exclusive events to a class example with 15 English, 10 history, and 5 both, showing the probability of taking either subject as 4/5.

Apply the addition rule for non-mutually exclusive events to a dice roll by identifying prime and odd outcomes, calculating their intersection, and obtaining a union probability of 2/3.

Explore the addition rule for non-mutually exclusive events by calculating probability of even numbers or perfect squares among cards 1 to 12, using union minus intersection to arrive at two-thirds.

Apply the addition rule for non-mutually exclusive events to find P(prime or multiple of five) among numbers 1 to 40, using inclusion-exclusion.

Learn the multiplication rule for probability, distinguishing independent and dependent events, with practical examples such as coin tosses, draws without replacement, dice rolls, and conditional probabilities.

Apply the multiplication rule for independent events to compute the probability of drawing red, then white, then black with replacement from 8 red, 6 white, and 4 black balls.

Apply the multiplication rule to independent events: with replacement, drawing two red balls from a box of two red and four blue yields probability (2/6) × (2/6) = 1/9.

Compute the probability of getting a two on one die and a five on the other using the multiplication rule for independent events, yielding 1/36.

Explore the multiplication rule for independent events by solving a probability exercise with replacement, calculating chances of both black, or one black and one red.

Compute the probability of rolling a sum of seven with two independent six-sided dice by counting the six favorable outcomes among 36 total possibilities, yielding 1/6.

Use the multiplication rule for independent events to compute the probability of exactly two heads in three tosses of a fair coin, illustrating three favorable outcomes out of eight.

Apply the multiplication rule for independent events to solve red and green balls with replacement and a die‑and‑coin scenario, yielding a 1/4 probability.

Apply the multiplication rule for dependent events to compute the probability that both drawn birds are red without replacement, using (4/9) × (3/8) = 1/6.

From five red and seven green, draw two without replacement and sum the two dependent scenarios: red first then green and green first then red, yielding 35/66.

Apply the multiplication rule for dependent events to compute the probability of selecting two defective bulbs from ten without replacement, yielding 3/10 and 2/9 for a final 1/15.

Apply the multiplication rule for dependent events to compute the probability of selecting two female teachers without replacement.

Explore conditional probability and dependent events by applying p(a|b) = p(a∩b)/p(b) to find the probability of A given that B has occurred.

Apply conditional probability to determine the probability that a non-blue draw is green, given five blue bars, three green bars, and two red bars, yielding 3/5.

Explore conditional probability by finding the probability a randomly chosen basketball player also plays soccer; compute P(soccer and basketball) over P(basketball), yielding 5/18.

Compute the conditional probability of Germany given a foreign language course. With 60% foreign language and 10% Germany within that group, the result is 10%.

Compute the probability that the second draw is blue given the first is red, drawing two balls without replacement from a box of four red and six blue balls.

Explore the theorem of total probability by partitioning a sample space into non-overlapping events that cover the space. Compute P(B) as the sum of P(B|A_i)P(A_i) across partitions.

Apply the theorem of total probability to two machines, using a sample-space partition and defect rates of 0.7, 0.3, 0.01, and 0.02 to obtain a 0.013 defect probability.

Apply the theorem of total probability to a three-level exam to compute the probability of a correct answer. Weight easy, medium, and hard questions by their proportions to get 0.795.

Use the total probability approach to compute the GPS likelihood for a random car: 0.6×0.9 + 0.4×0.7 = 0.82, combining electric and gas vehicle GPS rates.

Apply the theorem of total probability to the exercise, calculating the probability that a randomly chosen call is a complaint when the result is 90.5 percent (0.195).

Apply the theorem of total probability through the exercise, converting 85.5% to 0.855 and assessing correctness while reading and working through the question.

Apply the theorem of total probability to solve a practice question and verify your result, arriving at 2.097% and confirming correctness.

Solve this intuitive exercise on the theorem of total probability, illustrating how reaching 99.65 percent reflects correct application.

Practice the theorem of total probability with a straightforward exercise, read the question, work through it, and verify accuracy—81.4% indicates correct understanding.

Explore decision trees to solve probability questions, using a bag with yellow and white balls to build without-replacement and with-replacement scenarios, and examine independent versus dependent events.

Apply a decision tree to compute the probability of drawing two red balls without replacement from three red and five blue balls, deriving a total probability of 3/28.

Compute the probability of drawing a red marble, then a blue marble, without replacement, from six red and four blue marbles, using a decision tree: 6/10 × 4/9 = 4/15.

Explore the probability of drawing red then blue with replacement from six red and four blue marbles, using multiplication to obtain 6/25.

calculate the probability of drawing red, blue, then green without replacement from a bar containing 5 red, 4 blue, and 3 green marbles by multiplying 5/12, 4/11, and 3/10.

Explore probability with a two-flip coin: compute the chance of two heads, using independent events and conditional probability, showing that P(heads both) = 1/4.

Explore a decision tree exercise to calculate the probability of rolling a four on the die and getting a head on the coin, yielding 1/12.

Read the decision tree exercise, solve the displayed question, pause to work it out, and confirm that three over ten is correct.

Explains a decision tree exercise to compute the probability of drawing two balls of the same color with replacement from four red and five blue balls, yielding 41/81.

Explore a probability scenario with draws without replacement and compare it to the earlier version; verify correctness when the result is four over nine.

Compute the overall feasible bug probability using a decision tree by multiplying each cause probability by its feasibility and summing: 0.6×0.9 + 0.3×0.8 + 0.1×0.5 = 0.83.

Use a decision tree to compute the two-attempt pass probability, with first-attempt success 0.7 and second-attempt success 0.2 given failure, yielding 0.76.

Apply a decision tree using disease probabilities (0.5, 0.3, 0.2) and treatment effectiveness (0.7, 0.6, 0.8) to compute overall effectiveness.

Apply a decision tree approach to compute the probability a customer returns after winning a small, medium, or large prize by weighting each return rate by its prize probability.

Compute probability that an April flight takes off on time using a decision tree: weight sunny 75%, cloudy 20%, rainy 5% with on-time rates 95%, 85%, 60%, then sum.

Engage in a decision tree exercise by reading the question carefully, stopping after reading, and confirming you have the correct answer.

Evaluate a decision tree exercise that centers on a 0.54 result and validates the correct answer.