Exercises: Simple events

Write the sample space associated with the random experiment consisting of extracting 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 a piece is extracted at random, calculate the probability that it is a nut.

A fair cubic die with faces numbered from $1$ to $6$ is rolled. Determine the elements that form the complementary event to "rolling a number less than $3$" and calculate its probability.

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

In the experiment of rolling a $6$-sided die, classify the following events into 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. Indicate 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 elements 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 $3$.

Two fair cubic dice are rolled simultaneously and the sum of the scores obtained on the upper faces is recorded. Calculate the probability that the sum is exactly $8$.

In an electronic components factory, a quality control inspects a batch of $500$ pieces. After the inspection, $15$ defective pieces have been found. If a customer chooses a piece at random from that batch, what is the probability that it works correctly?

Knowing that the probability of an event $A$ occurring is $P(A) = \frac{9}{20}$, the probability of an 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 of at least one of the two events occurring.

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

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

In a class of $30$ students, $18$ practice swimming, $15$ practice tennis and $5$ do not practice either of the two sports. If a student is chosen at random, calculate the probability that they practice 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 tossed simultaneously. Describe the sample space and calculate the probability of obtaining exactly two heads.

In an urn there are $x$ red balls and $6$ white balls. If it is known that the probability of extracting 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 formed by three concentric circles of radii $10\text{ cm}$, $20\text{ cm}$ and $30\text{ cm}$. If a dart hits at random any point on the target (guaranteeing that it always hits inside the largest circle), calculate the probability that it lands in the intermediate circular ring (the area between the radius of $10\text{ cm}$ and that of $20\text{ cm}$).

Let $A$ and $B$ be two events of 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 complementary event of the intersection, that is, $P(\overline{A \cap B})$.

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

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

A survey of $200$ customers of a café 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 so that the probability of rolling an even face is twice the probability of rolling an odd face. Knowing that all even faces have the same probability as each other, and the same occurs with the odd faces, calculate the exact probability of rolling a $6$.

There are two mutually exclusive events $A$ and $B$ of the same random experiment. Knowing that the probability that $A$ does not occur is $P(\overline{A}) = 0.7$ and the probability that at least one of the two occurs is $P(A \cup B) = 0.8$, calculate the probability that event $B$ occurs.