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 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 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 $3$.

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.

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.