Exercises: Probabilities

Classify the following experiments as random or deterministic, briefly justifying your answer:

1. a) Observing whether it will rain in your city tomorrow.
2. b) Measuring the time it takes for a stone to fall from a second floor.
3. c) Recording the winning number in the next lottery draw.
4. d) Knowing the temperature at which distilled water boils at sea level.

You have the following wheel of fortune divided into eight equal sectors. If you spin it and record the resulting color:

1. a) Write the sample space $E$.
2. b) State two elementary events.
3. c) List the elements of the compound event $A$: "landing on a primary color" (considering red, blue and yellow as such).

In a sports bag there are $6$ soccer balls, $4$ basketballs and $2$ handballs. If we draw a ball at random to start training, calculate the probability of the following events applying Laplace's rule:

1. a) $P(\text{football})$
2. b) $P(\text{basketball})$
3. c) $P(\text{not a handball})$

A board game uses a dodecahedral die (12 equal faces) numbered from $1$ to $12$. Calculate the probability of obtaining:

1. a) A prime number.
2. b) A multiple of $4$.
3. c) A number greater than $9$.

We draw a card from a Spanish deck of $40$ cards. Find the probability of the following events:

1. a) Drawing the King of Cups.
2. b) Drawing an ace.
3. c) Drawing a card from the Coins suit.
4. d) Drawing a face card (Jack, Knight, or King).

A drawing pin has been dropped many times to observe whether it lands point up or point down. The results are collected in the following table:

Drops	$50$	$100$	$500$	$1000$
Point up	$18$	$32$	$155$	$312$

1. a) Calculate the relative frequency of the event "landing point up" for each column.
2. b) Based on the Law of Large Numbers, what value would you assign to the probability of the drawing pin landing point up?

A weather service announces that the probability of it snowing in the mountains tomorrow is $0.35$.

1. a) What is the probability that it does not snow?
2. b) If event $A$ is "it snows", what is the name of the "it does not snow" event relative to $A$?
3. c) Express both probabilities as a percentage.

We toss two $1$ peso coins into the air and observe the result on the upper faces.

1. a) Draw the tree diagram for this experiment.

1. b) Write the complete sample space.

1. c) What is the probability of getting exactly two «tails»?

You can do it on paper and send a photo with your answer.

In a playlist on a music streaming service, Lucas has 40 songs of different genres: 12 are Pop, 8 are Rock, 15 are Trap and 5 are Indie. If Lucas selects the "shuffle" option, calculate the probability that the first song played:

1. a) Is Trap.
2. b) Is not Indie.
3. c) Is Pop or Rock.

We roll a role-playing game die shaped like a dodecahedron (12 faces numbered from 1 to 12). We define the following events:

$A = \text{"Getting a multiple of 3"}$

$B = \text{"Getting a number greater than 7"}$

Calculate the following probabilities:

1. a) $P(A)$
2. b) $P(B)$
3. c) $P(A \cap B)$
4. d) $P(A \cup B)$

In a 3rd ESO class, a survey was conducted on the preferred social network for following influencers. The results are shown in the following table:

	TikTok	Instagram	TOTAL
Boys	12	6	18
Girls	10	12	22
TOTAL	22	18	40

If we choose a student at random, calculate the probability that they:

1. a) Are a girl.
2. b) Prefer Instagram.
3. c) Are a boy who prefers TikTok.
4. d) Prefer Instagram, given that they are a boy.

In a video game tournament, a player has a probability of $0.6$ of winning the first match. If they win the first, their confidence increases and the probability of winning the second is $0.7$. However, if they lose the first, the probability of winning the second drops to $0.3$.

Use a tree diagram to calculate the probability that they:

1. a) Win both matches.
2. b) Win at least one of the two matches.

In a sports bag there are 5 white T-shirts and 3 black T-shirts. If we draw two T-shirts at random and consecutively without returning the first to the bag, calculate the probability that:

1. a) Both are white.
2. b) They are of different colors.
3. c) The second is black, given that the first was white.

A student has found that the bus to school is late 4 out of every 10 days because of traffic. If to get to school he has to take two different buses that run independently, and each of them is late with that same probability, calculate:

1. a) The probability that both buses arrive on time.

1. b) The probability that at least one of the two buses is late.

You can use the following tree diagram, which you have to complete yourself with the probabilities of each branch:

A coin is tossed and a spinner divided into 5 equal sectors numbered from 1 to 5 is spun.

1. a) Write the sample space of this combined experiment.
2. b) What is the probability of getting "Heads" and an even number?
3. c) What is the probability of getting "Tails" and a number less than 5?

In a lottery drum there are 100 balls numbered from 00 to 99.

1. a) What is the probability that the drawn ball ends in 7?
2. b) What is the probability that it is a number between 20 and 35 (inclusive)?
3. c) What is the probability that it is neither an even number nor a multiple of 10?

A video game developer is testing a new reward system. They have programmed a virtual chest that, when opened, uses an algorithm to decide the prize. According to data from the last $2000$ chests opened by players, the following results were obtained:

• Common item: $1240$ times.

• Rare item: $680$ times.

• Legendary item: $80$ times.

1. a) Calculate the empirical probability (relative frequency) of obtaining a legendary item.
2. b) If the developer claims that the theoretical probability of obtaining a rare or legendary item is $40\%$, do you think the algorithm is working correctly according to the Law of Large Numbers? Reason your answer by comparing the probabilities.
3. c) If a player opens two chests in a row, what is the probability that both contain a common item? (Assume that the events are independent).

Two esports teams, the "Pixels" and the "Lag-Killers", play the final of a best-of-three tournament (the first to win two matches wins). From previous statistics, we know that in each individual match, the "Pixels" have a probability of winning of $P(P) = 0.6$, while the "Lag-Killers" have $P(L) = 0.4$. There is no possibility of a draw.

Use a tree diagram to calculate:

1. a) The probability that the "Pixels" win the tournament in just two matches.
2. b) The probability that the tournament is decided in the third match.
3. c) The total probability that the "Lag-Killers" come back to win the tournament.

In a school reforestation project, we have two planters with seeds. Planter $A$ contains $4$ oak seeds and $6$ pine seeds. Planter $B$ contains $5$ oak seeds and $3$ pine seeds.

We carry out the following experiment: we draw a seed at random from planter $A$ and transfer it to planter $B$ without looking at it. Then, we draw a seed from planter $B$.

1. a) What is the probability that the seed drawn from $B$ is an oak seed?
2. b) What is the probability that both seeds drawn (the one transferred from $A$ to $B$ and the one from $B$) are of the same species?

A courier company uses drones to deliver packages in a rural area. The landing area is a square with side length $10$ meters. Due to inaccuracy in the wind sensor, the package can land at any point in the square with equal probability. Inside this square, a safety circle with a radius of $3$ meters has been painted. If the package lands outside the circle, the drone must land again for safety.

1. a) Calculate the probability that a package lands inside the safety circle on the first attempt.
2. b) If the drone makes two independent deliveries, what is the probability that at least one of them lands outside the circle? (Use $\pi \approx 3.14$).

In a board game, two six-sided cubic dice are rolled. A player wins if the positive difference between the scores on the two dice is a prime number.

1. a) Construct a two-way table to show all possible differences.
2. b) What is the probability of winning the game?
3. c) Is it more likely to get a difference that is a prime number or a difference that is a multiple of $3$? Justify your answer by calculating both probabilities.

You can do it on paper and send a photo with your answer.

A study on social media use was carried out in a high school with $300$ students. The results are as follows:

• $180$ students use Instagram.

• $130$ students use TikTok.

• $40$ students do not use either social network.

If we choose a student at random, calculate the probability that they:

1. a) Use both social networks.
2. b) Use Instagram but not TikTok.
3. c) Also use Instagram, given that the student uses TikTok.

A multiple-choice exam consists of $4$ questions with $3$ response options each, of which only one is correct. A student who has not studied decides to answer all the questions at random.

1. a) What is the probability that they do not get any question correct?
2. b) What is the probability that they get exactly two questions correct?
3. c) If they need to get at least $3$ questions correct to pass, what is their probability of passing the exam?

In a music box there are $12$ guitar picks: $5$ are red, $4$ are blue and $3$ are black. If we draw three picks from the box, one after another and without replacement, calculate:

1. a) The probability that all three picks are the same color.
2. b) The probability that the first is red, the second is blue and the third is black, in that exact order.
3. c) The probability that at least one of the drawn picks is black.