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 colour 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.

In a box there are $6$ blue pens and $4$ black pens. We draw two pens consecutively and without replacement. We want to calculate the probability that both pens drawn are the same color.

Sergio solves it as follows:

«Since there are two possible colors, first I calculate the probability of two blue pens: $\frac{6}{10} \cdot \frac{6}{10} = \frac{36}{100}$. Then that of two black pens: $\frac{4}{10} \cdot \frac{4}{10} = \frac{16}{100}$. Finally, since either of the two cases works for me, I multiply both results: $\frac{36}{100} \cdot \frac{16}{100} = 0,0576$, that is, $5,76\,\%$».

1. a) Identify and explain the two conceptual errors Sergio has made: one concerning the dependence between consecutive stages and the other concerning the combination of the final branches.
2. b) Correct the approach and calculate the exact probability of obtaining two pens of the same color, as an irreducible fraction and as a percentage.

In a board game, to overcome an obstacle you must roll a six-sided die twice in a row. The rule states that you overcome the obstacle if you get at least one number greater than $4$ (that is, a $5$ or a $6$) on either of the two rolls.

1. a) Estimate: Before performing any exact mathematical calculation, reflect and answer: do you think the final probability of overcoming the challenge will be less than $50\,\%$, exactly $50\,\%$, or greater than $50\,\%$? Write a sentence justifying your intuition.
2. b) Check: Model the double roll as a compound event. First calculate the probability of the complementary event ("not getting more than 4 on any roll") by applying the multiplication rule, and, from it, find the actual probability of overcoming the obstacle as an irreducible fraction and a percentage.
3. c) Compare the result with your initial estimate: did it match what you had thought? Reason why linking two chances makes the overall cumulative probability noticeably greater than that of a single individual roll (which is $\frac{2}{6} = \frac{1}{3} \approx 33,3\,\%$).

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 centre, 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 colour, 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 precision marksman has a fixed probability of $0,7$ of hitting a target on each shot. If they take $3$ completely independent shots, we want to calculate the probability that they hit the target at least once.

To solve this problem, two different strategies can be proposed:

• Method A (direct): Identify all favorable outcomes (hitting exactly once, hitting exactly twice, or hitting all three shots), calculate the probability of each branch of the tree, and add them together.

• Method B (complementary event): Calculate the probability of missing all three shots consecutively and subtract that value from $1$.

1. a) Compare both approaches: how many branches of the tree diagram would you have to calculate and add up with Method A compared to the operations needed with Method B? Justify which one is more efficient and safer against algebraic slips.
2. b) Apply the most convenient method to calculate the requested probability, expressing the final result in decimal format and as a percentage.
3. c) If the marksman were to take $5$ shots instead of $3$, reason how the amount of calculations required in Method A would change compared to Method B.

According to the records of a provincial traffic department in Spain concerning candidates obtaining a category B driving licence:

• The probability that a student passes the theory test on their first attempt is $70\,\%$.

• Among those who pass the theory test on the first attempt, the probability of passing the practical test on the first attempt is $60\,\%$.

• Among those who fail the theory test on the first attempt (and pass it on subsequent attempts), the probability of passing the practical test on the first attempt is $40\,\%$.

1. a) Model the process by describing the stages of the compound experiment and the probabilities assigned to each branch.
2. b) If a candidate is chosen at random, calculate the probability that they pass both the theory and practical tests on their first attempt.
3. c) Calculate the total probability that a randomly chosen candidate passes the practical test on their first attempt, regardless of whether they passed the theory test on the first attempt or needed more attempts. Explain why it is necessary to add the probabilities of two different paths of the tree.

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.

In a drawer there are exactly $10$ socks stored, identical to the touch: $n$ of them are blue and the remaining $10 - n$ are grey. A person pulls out two socks consecutively in the dark without replacement.

Knowing that the probability that the two drawn socks are of different colors is exactly equal to $\frac{8}{15}$:

1. a) Model the experiment as a compound event of two consecutive stages and write the algebraic expression for this probability in terms of $n$.
2. b) Set up an equation using the known data and solve algebraically for the value of $n$ to determine how many blue socks there can be in the drawer.
3. c) Interpret the fact of obtaining two different mathematical solutions: why does it make complete logical sense in the physical context of the problem that there are two valid values for $n$?

A museum protects a valuable sculpture using a security system made up of two independent sensors: an infrared perimeter barrier (Sensor A) and a pressure plate under the plinth (Sensor B). In the event of a real intrusion, Sensor A triggers with a probability of $0,90$, while Sensor B triggers with a probability of $0,85$.

The security company proposes two possible technical configurations to trigger the main alarm:

• Parallel configuration (redundant): The alarm sounds if at least one of the two sensors is triggered.

• Series configuration (restrictive): The alarm sounds only if both sensors are triggered simultaneously (designed to minimize false alarms caused by drafts or dust).

1. a) Model the parallel configuration using compound events and calculate the probability that the alarm sounds if an intruder enters (use the multiplication rule with complementary events).
2. b) Model the series configuration and calculate its activation probability in the event of an intrusion.
3. c) The museum management is hesitating between keeping the parallel configuration or replacing the entire system with a single state-of-the-art laser sensor whose individual detection reliability is $0,96$. Based on your previous calculations, mathematically justify which option provides greater actual protection against theft and argue what operational drawback the more sensitive system presents regarding false alarms.