Exercises: Simple events

Write down the sample space associated with the random experiment of drawing a ball from an urn containing $5$ balls numbered from $1$ to $5$.

In a toolbox there are $12$ screws, $8$ nuts and $5$ nails. If an item is picked at random, calculate the probability that it is a nut.

A fair six-sided die with faces numbered from $1$ to $6$ is rolled. Determine the outcomes that form the complement of the event "rolling a number less than $3$" and calculate its probability.

A card is drawn at random from a deck of $40$ cards that includes knights. Calculate the probability of obtaining a knight.

In the experiment of rolling a $6$-sided die, classify the following events as elementary, compound, certain or impossible:

• $A = $ "rolling a $4$"

• $B = $ "rolling a number greater than $6$"

• $C = $ "rolling an even number"

• $D = $ "rolling a number less than $10$"

Consider the experiment of drawing a card from a Spanish deck of $40$ cards. State whether the events $A = $ "drawing a face card" and $B = $ "drawing a coin" are compatible or incompatible. Justify your answer.

Given the sample space $E = \{1, 2, 3, 4, 5, 6, 7, 8\}$ and the events $A = \{2, 4, 6, 8\}$ and $B = \{1, 2, 3, 4\}$, determine the outcomes that form the events $A \cup B$ and $A \cap B$.

A fairground spinner is divided into $8$ equal circular sectors, numbered from $1$ to $8$. Calculate the probability that, when spun, the arrow stops on a multiple of $3$.

Seán tosses two fair $1$-euro coins simultaneously and claims the following: "The sample space of this experiment is $E = \{ \text{two heads}, \text{one head and one tail}, \text{two tails} \}$. Since there are $3$ possible outcomes and only one of them is 'obtaining one head and one tail', Laplace's rule states that the probability of getting one head and one tail is $\frac{1}{3}$". a) Explain with reasoning what Seán's conceptual error is and why it is not valid to apply Laplace's rule to the sample space he proposed. b) Write down the correct sample space formed by equally likely elementary events and calculate the actual probability of the event analysed by Seán.

In a Spanish-style deck of $40$ cards (a traditional deck used in Spain) (divided into $4$ suits of $10$ cards each: coins, cups, swords, and batons, numbered from $1$ to $7$ plus jack, horse, and king), a card is drawn at random. Consider event $A$: "draw a card that is a face card (jack, horse, or king) or has a prime number ($2$, $3$, $5$, or $7$)". a) Before calculating, reasonably estimate whether $P(A)$ will be greater than, less than, or equal to $0.5$, by analysing how many cards of a single suit meet at least one of the two conditions. b) Check your estimate by calculating the exact value of $P(A)$ using Laplace's rule (express the result as an irreducible fraction and as a decimal number) and determine whether your initial prediction was correct.

Two fair six-sided dice are rolled simultaneously, and the sum of the scores on the top faces is recorded. Calculate the probability that the sum is exactly $8$.

In an electronic component factory, quality control inspects a batch of $500$ parts. After the inspection, $15$ defective parts are found. If a customer chooses a part at random from that batch, what is the probability that it works correctly?

Given that the probability of event $A$ occurring is $P(A) = \frac{9}{20}$, the probability of event $B$ occurring is $P(B) = \frac{3}{10}$, and the probability of both occurring at the same time is $P(A \cap B) = \frac{1}{10}$, calculate the probability that at least one of the two events occurs.

Counters with each of the letters of the word $ESPECTACULO$ are placed in an urn. If a counter is drawn at random, calculate the probability that the drawn letter is a vowel or the letter $C$.

A basketball player has taken $80$ free throws during the season, scoring $64$ of them. Using relative frequency as an estimate of probability, what is the probability that he misses his next free throw?

In a class of $30$ students, $18$ do swimming, $15$ play tennis, and $5$ do neither sport. If a student is chosen at random, calculate the probability that they do both sports simultaneously.

In an animal shelter, $40\%$ of the dogs are large, $35\%$ are medium-sized, and the rest are small. If a dog is selected at random for a walk, calculate the probability that it is not medium-sized.

Three fair coins are flipped simultaneously. Describe the sample space and calculate the probability of obtaining exactly two heads.

You are designing a circular spinner for a school fair divided into continuous sectors of three colours: red, blue, and green. To adjust the prizes for the game, the design must satisfy the following conditions: 1) The probability that the spinner does not land on green is exactly $\frac{7}{10}$. 2) The event "landing on red" has a probability that exceeds that of the event "landing on blue" by $0.1$. Determine the probability of each of the three colours (elementary events) and calculate the angle in degrees that the sector of each colour must span in the full $360^\circ$ circle. Justify your steps using the complement rule and the rule for the union of mutually exclusive events.

In a secondary school, the head of studies analyses the enrolment of the $120$ Year 10 (4.º de ESO) students in the elective pathway subjects. Official data show that $72$ students take Biology and Geology, $48$ take Physics and Chemistry, and $24$ take both subjects simultaneously. If a Year 10 student from this school is selected at random: a) Model the situation by defining the elementary events and calculate the probabilities $P(B)$, $P(F)$, and $P(B \cap F)$. b) Apply the formula for the probability of the union to calculate the probability that the student takes at least one of the two subjects. c) Using the complement rule, calculate the probability that the student takes neither of the two subjects.

In an urn there are $x$ red balls and $6$ white balls. Knowing that the probability of drawing a red ball at random is $\frac{2}{5}$, set up an equation and determine the total number of balls in the urn.

A circular target is made of three concentric circles with radii $10\text{ cm}$, $20\text{ cm}$ and $30\text{ cm}$. If a dart hits any point on the target at random (guaranteeing it always lands inside the outer circle), calculate the probability that it lands in the middle annulus (the area between the radius of $10\text{ cm}$ and that of $20\text{ cm}$).

Let $A$ and $B$ be two events in a random experiment such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$ and $P(A \cup B) = \frac{3}{4}$. Using the properties of probability, calculate the probability of the complement of the intersection, that is, $P(\overline{A \cap B})$.

To open a safe, a $3$-digit numerical code must be entered (from $000$ to $999$). A burglar who does not know the passcode tries a random number. Calculate the probability that the entered code has all three digits identical.

In a draw, a ball is drawn from a drum containing $100$ balls numbered from $1$ to $100$. Calculate the probability that the number on the drawn ball contains at least one digit $7$.

A survey of $200$ coffee shop customers reveals that $120$ have coffee, $90$ have toast, and $40$ have neither. If a customer is chosen at random, calculate the probability that the customer has coffee but does NOT have toast.

A six-sided die is biased such that the probability of rolling an even number is twice the probability of rolling an odd number. Given that all even faces are equally likely and all odd faces are equally likely, calculate the exact probability of rolling a $6$.

Two incompatible events $A$ and $B$ from the same random experiment are given. Knowing that the probability of $A$ not occurring is $P(\overline{A}) = 0.7$ and the probability of at least one of the two occurring is $P(A \cup B) = 0.8$, calculate the probability of event $B$ occurring.

Three fair cubic dice with faces numbered from $1$ to $6$ are rolled simultaneously. The goal is to calculate the probability of event $A$: "obtain at least one $5$ in the roll". Two students discuss which approach to follow: Strategy 1 (direct method) consists of decomposing $A$ into the sum of three mutually exclusive events ("exactly one $5$", "exactly two $5$s", and "three fives") and adding their probabilities; Strategy 2 (complementary event) consists of calculating the probability of $\bar{A}$ ("obtain no $5$ on any of the three dice") and applying $P(A) = 1 - P(\bar{A})$. a) Fully develop both methods to obtain the result as an irreducible fraction, checking that they match. b) Justify which of the two strategies is more efficient and explain how the choice of method would be affected if the experiment consisted of rolling $8$ fair dice instead of $3$.

In a Mathematics class, a random experiment with two events, $A$ and $B$, from the same sample space is analysed, whose probabilities are $P(A) = 0.65$ and $P(B) = 0.55$. Aoife states: 1) "Events $A$ and $B$ can be mutually exclusive if we design the experiment well". 2) "The minimum possible value for the probability of the intersection $P(A \cap B)$ is $0$". 3) "The probability of the union $P(A \cup B)$ could be $1.2$ if we add their probabilities". a) Reasonably refute Aoife's three statements by applying the axioms and properties of probability ($P(A \cup B) \le 1$ and the addition rule). b) Deduce the exact interval $[p_{\min}, p_{\max}]$ in which $P(A \cap B)$ must necessarily lie. c) Design a specific example using an urn with $20$ balls numbered from $1$ to $20$ where $P(A) = 0.65$ and $P(B) = 0.55$ are exactly satisfied, and the minimum value $p_{\min}$ of the intersection is attained.