Exercises: Compound events

We flip two coins independently. Calculate the probability of obtaining exactly two tails.

In a box there are $5$ red balls and $3$ blue balls. A ball is drawn, its color recorded, replaced in the box, and a second ball is drawn. Find the probability that both balls are red.

We roll a fair six-sided die (with faces numbered from $1$ to $6$) and flip a coin. What is the probability of obtaining an even number on the die and a head on the coin?

From a Spanish deck of $40$ cards, two cards are drawn consecutively without replacement. Calculate the probability that both cards are Kings.

In a restaurant, $60\%$ of customers order the set menu and $40\%$ order à la carte. Of those ordering the set menu, $70\%$ have dessert. Of those ordering à la carte, $50\%$ have dessert. Calculate the probability that a customer chosen at random orders the set menu and has dessert.

In a Year 11 class there are $12$ boys and $18$ girls. It is known that $4$ boys and $6$ girls wear glasses. If we choose a student at random, what is the probability that they are a girl who wears glasses?

We spin the two independent spinners shown in the figure. Calculate the probability that the first spinner lands on the number $3$ and the second spinner lands on green (V).

On a street there are two independent traffic lights. The probability of the first one being green is $0.4$ and the probability of the second one being green is $0.5$. Find the probability of encountering both traffic lights on green when travelling down that street.

Three cards are drawn consecutively and without replacement from a Spanish deck of $40$ cards. Calculate the probability of obtaining three cards of the suit of coins.

At a sports center, tennis and padel classes are offered. There are $120$ members enrolled in these classes, and each member plays only one sport. It is known that $70$ members play padel and that there are $50$ women in total. Furthermore, $30$ women play tennis. If a member is chosen at random, calculate the probability that they are a man who plays padel.

We have two urns. Urn $A$ contains $4$ white balls and $2$ black balls. Urn $B$ contains $3$ white balls and $5$ black balls. We flip a fair coin; if it lands on heads, we draw a ball from urn $A$, and if it lands on tails, we draw a ball from urn $B$. Find the probability that the drawn ball is white.

An electronic component factory has two machines. Machine $M_1$ produces $60\%$ of the components and machine $M_2$ the remaining $40\%$. It is known that $2\%$ of the components produced by $M_1$ are defective, while $5\%$ of those produced by $M_2$ are defective. If a component is selected at random from the warehouse, what is the probability that it is defective?

Two basketball players, Marta and Laura, each take a free throw. The probability that Marta scores is $0.8$ and the probability that Laura scores is $0.7$. Their shots are independent events. Calculate the probability that at least one of them scores her free throw.

In a game, a participant first draws a card from a Spanish deck of $40$ cards. If the drawn card is of the suit of cups, he rolls a fair six-sided die (from $1$ to $6$). If the card is NOT of cups, he flips a coin. What is the probability that the participant ends up getting a $5$ on the die?

We have two boxes. Box 1 contains $3$ red counters and $2$ green counters. Box 2 contains $1$ red counter and $4$ green counters. A counter is drawn at random from Box 1 and, without looking at its color, is placed into Box 2. Next, a counter is drawn at random from Box 2. Calculate the probability that this second counter is red.

An archer has a probability of $0.85$ of hitting the bullseye on each shot. If he takes three consecutive independent shots, find the probability that he misses all three shots.

A medical test to detect a disease has the following characteristics: if a person is ill, the test gives a positive result in $95\%$ of cases. If the person is healthy, the test gives a positive result (false positive) in $3\%$ of cases. It is known that $2\%$ of the population suffers from this disease. If a person is chosen at random and tested, what is the probability that the result is positive?

Two e-sports teams, Alpha and Beta, play the final of a best-of-three tournament (the first to win two games becomes champion). The probability of Alpha winning a game is $0.6$ and that of Beta winning is $0.4$ (there are no draws). The game results are independent. Calculate the probability that team Alpha wins the tournament.

An urn contains $x$ white balls and $5$ black balls. Two balls are drawn consecutively without replacement. Knowing that the probability of both balls being white is $\frac{1}{6}$, set up a quadratic equation and determine the initial number of white balls $x$ that were in the urn.

The following diagram shows a pipe system with three valves $A$, $B$ and $C$. Water flows from left to right. For water to reach the end, it must pass through valve $A$ or through the lower branch containing valves $B$ and $C$ in series. The probabilities of the valves being open are $P(A) = 0.7$, $P(B) = 0.8$ and $P(C) = 0.9$. Knowing that the operation of each valve is independent, calculate the probability that water successfully passes through the system.

At an international conference, $40\%$ of the attendees are European and $60\%$ are American. It is known that $30\%$ of the Europeans and $20\%$ of the Americans are bilingual. If we select two attendees at random with replacement, what is the probability that exactly one of the two is bilingual?

Cards are drawn one by one without replacement from a Spanish deck of $40$ cards until the first Ace appears. The experiment ends as soon as an Ace is drawn or when a maximum of $3$ cards have been drawn. Calculate the probability that the experiment ends having found an Ace.

A dart is thrown at the target in the figure, formed by two concentric circles of radii $R$ and $2R$. The dart always hits the target, and the probability of hitting a region is proportional to its area. If two darts are thrown independently, calculate the probability that one dart lands in the central region (A) and the other in the outer region (B).

We have two urns. Urn $U_1$ contains $2$ red balls and $2$ black balls. Urn $U_2$ is initially empty. Two balls are drawn simultaneously from $U_1$ and placed into $U_2$. Next, a ball is drawn at random from $U_2$. Calculate the probability that the ball drawn from $U_2$ is red.