Exercises: Compound events

We toss 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 extracted, its color is recorded, it is returned to the box and a second ball is extracted. Find the probability that both balls are red.

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

Two cards are drawn consecutively without replacement from a Spanish deck of $40$ cards. Calculate the probability that both cards are Kings.

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

In a 10th grade 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 it is 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 the color green (V).

In a street there are two independent traffic lights. The probability that the first one is green is $0,4$ and the probability that the second one is green is $0,5$. Find the probability of finding both traffic lights green when passing through 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 coin suit.

A sports center offers tennis and padel classes. There are $120$ members enrolled in these classes, and each member practices 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 it is 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 toss a fair coin; if it lands on heads, we extract a ball from urn $A$, and if it lands on tails, we extract a ball from urn $B$. Find the probability that the extracted ball is white.

An electronic components 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 in the warehouse, what is the probability that it is defective?

Two basketball players, Emma and Chloe, each take a free throw. The probability that Emma scores is $0,8$ and the probability that Chloe scores is $0,7$. Their throws are independent events. Calculate the probability that at least one of the two 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 from the cup suit, they roll a fair cubic die (from $1$ to $6$). If the card is NOT from the cup suit, they toss a coin. What is the probability that the participant ends up obtaining a $5$ on the die?

We have two boxes. Box 1 contains $3$ red tokens and $2$ green tokens. Box 2 contains $1$ red token and $4$ green tokens. A token is extracted at random from box 1 and, without looking at its color, it is placed into box 2. Then, a token is extracted at random from box 2. Calculate the probability that this second token is red.

An archer has a probability of $0,85$ of hitting the center of the target on each shot. If he takes three consecutive and 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 is positive in $95\%$ of cases. If the person is healthy, the test is positive (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 esports teams, Alpha and Beta, play the final of a best-of-three tournament (the first to win two games is crowned champion). The probability that Alpha wins a game is $0,6$ and that Beta wins is $0,4$ (there are no draws). The results of the games are independent. Calculate the probability that team Alpha wins the tournament.

An urn contains $x$ white balls and $5$ black balls. Two balls are extracted consecutively and without replacement. If we know that the probability that both balls are 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 $C$. The water flows from left to right. For the 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 that the valves are 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 the water manages to flow 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 from a Spanish deck of $40$ cards one by one and without replacement, until the first Ace appears. The experiment ends the moment 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 zone (A) and the other in the outer zone (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 $U_2$. Then, a ball is drawn at random from $U_2$. Calculate the probability that the ball drawn from $U_2$ is red.