Counting techniques

A restaurant offers $4$ starters, $3$ main courses and $2$ desserts. If a customer chooses their menu at random, what is the probability that they choose exactly the salad for starter, the salmon for main course and the cake for dessert?

A fair $6$-sided die is rolled $4$ consecutive times. What is the probability that the number $5$ is obtained on all rolls?

Four friends board a bus that has $6$ stops remaining until the end of the route. If each friend is equally likely to get off at any of the stops, what is the probability that all four friends get off at different stops?

In Ireland, the Road Safety Authority (RSA) assigns each vehicle a license plate consisting of a block of $4$ digits (from $0000$ to $9999$) followed by a block of $3$ consonant letters. To avoid offensive words, the $5$ vowels are excluded, and to avoid reading confusion, Ñ and Q are discarded, leaving exactly $20$ consonants available.\nA newly manufactured vehicle receives a license plate assigned randomly and equiprobably among all the combinations allowed by the system.\na) Model the sample space and calculate how many distinct theoretical license plates can be generated if repetition of digits and letters is allowed.\nb) A number enthusiast looks for a «symmetric» license plate: the $4$ digits must form a palindrome number (for example, $4884$ or $0330$) and the $3$ letters must form a palindrome (the first and third letters must match, as in $KPK$). What is the probability that a car is randomly assigned a symmetric license plate? Express the result as an irreducible fraction and as a percentage.

In a $100$-metre sprint, $6$ athletes take part. If all have the same probability of winning and there are no ties, what is the probability that the finishing order is exactly A, B, C, D, E, F?

In a chess club with $10$ members, a president, a secretary and a treasurer are to be chosen at random. What is the probability that Aoife is chosen as president, Seán as secretary and Niamh as treasurer?

All possible words, whether meaningful or not, are formed by rearranging the letters of the word "ROMA". If one of these words is chosen at random, what is the probability that it starts with the letter "M"?

Numbers with $4$ distinct digits are formed at random using the digits $1, 2, 3, 4, 5, 6$. What is the probability that the number formed is even?

From a group of $8$ students, $3$ are chosen at random to form a representative committee. What is the probability that the chosen ones are exactly the $3$ oldest students in the group?

In a box there are $5$ red balls and $4$ blue balls. $2$ balls are drawn at random simultaneously. What is the probability that both balls are red?

$5$ fair coins are tossed simultaneously. What is the probability of getting exactly $2$ heads and $3$ tails?

To represent your 5th Year group (consisting of $20$ students) before the school's Student Council, a committee of $3$ people must be chosen by lottery without distinction of roles or hierarchy. Cian states in class: «Since there are $20$ options for the first person, $19$ for the second, and $18$ for the third, the total number of possible committees is $20 \times 19 \times 18 = 6840$. Therefore, the probability that the chosen committee consists exactly of Aoife, Darragh, and me is $\frac{1}{6840}$».\na) Identify the conceptual error that Cian makes and explain why his calculation of the number of possible committees is not correct.\nb) Correct his reasoning by properly calculating the number of possible committees using counting techniques, and obtain the true probability that this specific trio is chosen.

In a tech company, $7$ engineers and $5$ economists are employed. A project team of $4$ people is to be chosen at random. What is the probability that the team consists of exactly $2$ engineers and $2$ economists?

From a deck of $40$ cards (four suits of ten cards each), $3$ cards are drawn at random without replacement. What is the probability that all three cards are of the same suit?

A batch of $20$ microchips contains $4$ defective ones. A random sample of $3$ microchips is selected for inspection. What is the probability that there is at least one defective microchip in the sample?

A multiple-choice exam consists of $10$ questions, each with $4$ answer options, only one of which is correct. If a student answers all questions at random, what is the probability that they get exactly $6$ questions right?

In a secondary school, $2$ places are announced for a summer language stay among 5th Year students. There are $5$ applicants from the Humanities and Social Sciences track and a group of $n$ applicants from the Science track ($n \ge 2$). The two places are awarded by consecutive lottery without replacement among all applications.\nThe head of studies announces that the probability that at least one of the two places goes to a Humanities and Social Sciences student is exactly equal to $\frac{7}{9}$.\na) Set up an equation relating $n$ to the given probability, using combinatorial counting techniques and relying on the complementary event (no places for Humanities and Social Sciences).\nb) Solve the resulting equation with full reasoning, justify the discarding of any invalid algebraic solutions, and conclude how many Science students and how many total students participated in the draw.

To form an evaluation committee of $5$ members in an educational center, its members must be chosen by lottery from a staff of $12$ available teachers: $7$ belong to the scientific-technological area and $5$ to the sociolinguistic area. The leadership requires that the committee be mixed (that is, it must necessarily include at least one teacher from each of the two branches). To find the probability that a random selection meets this requirement, three teachers propose different resolution methods:\n- Cian's method: ensures the presence of both branches by first choosing $1$ from Science ($7$ options) and $1$ from Humanities ($5$ options), and then choosing the remaining $3$ members from the $10$ remaining teachers, dividing the result by $\binom{12}{5}$.\n- Aoife's method: breaks down the favorable cases by adding the four disjoint valid configurations: $(1\text{ C, } 4\text{ H}) + (2\text{ C, } 3\text{ H}) + (3\text{ C, } 2\text{ H}) + (4\text{ C, } 1\text{ H})$, and divides the sum by $\binom{12}{5}$.\n- Seán's method: calculates the probability of the complementary event (that all are from Science or all are from Humanities) and subtracts it from $1$.\nYou are asked to:\na) Analyze Cian's proposal: prove numerically why it is incorrect and explain precisely what counting error his approach makes.\nb) Compare Aoife's and Seán's methods: justify which of the two is more efficient in terms of calculation and apply it to obtain the exact probability as an irreducible fraction.

$5$ mathematics books and $3$ physics books are to be placed at random on a shelf. What is the probability that all the physics books are placed together?

All possible words are formed by rearranging the letters of the word "BANANA". What is the probability that the three letters "A" appear together consecutively?

Four people leave their coats in a theatre cloakroom. Upon leaving, due to a power cut, the attendant hands back the four coats at random. What is the probability that exactly one person receives their own coat?

Eight people, including two siblings, sit at random around a circular table. What is the probability that the two siblings do not sit in adjacent seats?

$3$ distinct vertices of a regular decagon are chosen at random. What is the probability that the triangle formed by these $3$ vertices is a right-angled triangle?

A university department consists of $6$ tenured professors and $8$ associate professors. A committee of $5$ people is chosen at random. What is the probability that the committee has a majority of tenured professors (that is, $3$ or more)?

On the ground floor of a $8$-storey building, $5$ people get into the lift. If each person chooses at random the floor at which they get off (from $1$ to $8$), what is the probability that exactly two people get off on the same floor and the other three get off on floors that are different from each other and different from that of the pair?

A user forgets the $4$-digit PIN code of their mobile phone's SIM card. They remember with complete certainty two conditions: all four digits were distinct, and all were even digits (that is, chosen exclusively from $\{0, 2, 4, 6, 8\}$). They decide to try a random combination that satisfies both conditions.\na) Before carrying out any formal mathematical calculation, make an estimate: do you think the probability of unlocking the phone on the first try will be greater than, less than, or approximately equal to $1\,\%$? Briefly justify the intuition behind your estimate.\nb) Model the sample space using the multiplication principle or permutations without repetition to calculate the exact number of possible codes and the true probability of guessing it right on the first try (expressed as an irreducible fraction and as a percentage).\nc) Compare the exact value with your previous estimate and assess whether our intuition tends to underestimate or overestimate the effect of the multiplication principle.

In the emergency department of a public hospital, the night shift team every Saturday consists of $4$ doctors drawn by lottery from an available staff of $10$ professionals: $4$ attending physicians (veterans) and $6$ non-consultant hospital doctors (NCHDs, interns/registrars). For the shift to be operational according to the center's protocol, two conditions must be satisfied simultaneously:\n1. Safety care criterion: there must be at least $1$ attending physician on the team.\n2. Educational tutoring criterion: two specific residents, Aoife and Cian, follow a joint training plan, so they must necessarily do the shift together if either of them is called (that is, either both join the team or neither joins).\nIf the team of $4$ professionals is drawn equiprobably from the $10$ available:\na) Model the space of favorable cases by distinguishing the two disjoint scenarios imposed by Aoife and Cian's educational criterion, and calculate how many teams satisfy both criteria simultaneously.\nb) Determine the probability that the randomly selected team satisfies both regulatory requirements.\nc) Assess the organizational impact: calculate what the probability of obtaining a valid shift would be if only the care criterion were required (without the restriction on Aoife and Cian) and reflect on the cost in flexibility that block-grouping conditions entail.

From a standard deck of $52$ cards (consisting of $13$ different ranks across $4$ suits), $5$ cards are drawn at random. What is the probability of getting a full house (that is, three cards of one rank and two cards of another rank)?

$5$ distinguishable balls are distributed at random into $3$ distinguishable urns. What is the probability that none of the three urns is left empty?

A fair $6$-sided die is rolled $6$ consecutive times. What is the probability that all numbers from $1$ to $6$ appear exactly once, given that the first three rolls resulted exclusively in even numbers?

Unit challenge. Is your environment vulnerable to a brute-force attack? You are going to evaluate the robustness of common passwords by preparing a cybersecurity technical report.

1. Data collection: Conduct a brief anonymous survey of $10$ people (family or friends) or consult NCSC/CSO statistics to record which structures they use ($4$-digit PIN, $6$-letter lowercase passwords, or $8$-character alphanumeric passwords).

2. Counting techniques: Model each password format by identifying the sample space $\Omega$. Apply counting techniques (permutations with and without repetition, combinations) to calculate how many different passwords can be formed in each category.

3. Probability analysis: In a spreadsheet, calculate the probability according to Laplace of cracking the password on the first attempt and within $100$ attempts (without repeating passwords already tried). Define and illustrate compatible, incompatible, and complementary events (such as "not including numbers" versus "including at least one number").

4. Reasoned proposal: Write a one-page report relating the size of the sample space to the probability of vulnerability, proposing security guidelines based on your calculations.

Assessment criteria:

• Choice and correct calculation of appropriate counting techniques for each case.

• Rigour in the use of formal probability notation ($P(A)$, events, $\Omega$) and spreadsheet.

• Coherent justification of mathematical recommendations in the final report.