Exercises: Probability

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

1. a) Observing whether it will rain tomorrow in your city.
2. b) Measuring the time it takes for a stone to fall from a second floor.
3. c) Noting down 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 note the resulting color:

1. a) Write down the sample space $E$.
2. b) State two elementary events.
3. c) Write down 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 take out 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 dodecahedron-shaped die (12 equal faces) numbered from $1$ to $12$. Calculate the probability of getting:

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 suit of Coins.
4. d) Drawing a "face card" (Jack, Knight or King).

A pushpin has been dropped many times to observe whether it lands point up or point down. The results are recorded 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 pushpin 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 event of it not snowing called with respect to $A$?
3. c) Express both probabilities as a percentage.

We toss two $\$1$ coins in the air and observe the result of the upper faces.

1. a) Draw the tree diagram for this experiment.
2. b) Write down the complete sample space.
3. c) What is the probability of getting exactly two "tails"?

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

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

We roll a role-playing game die in the shape of 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 10th grade class, a survey was carried out 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) Given that they are a boy, prefer Instagram.

In a video game tournament, a player has a $0,6$ probability of winning the first match. If they win the first one, their confidence increases and the probability of winning the second is $0,7$. However, if they lose the first one, the probability of winning the second drops to $0,3$.

Use a tree diagram to help you 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 shirts and 3 black shirts. If we draw two shirts at random and consecutively without returning the first one to the bag, calculate the probability that:

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

A student has found that the bus to school arrives late 4 out of 10 days due to traffic. If they have to catch two different buses that operate independently to get to school, calculate:

1. a) The probability that both buses arrive on time.
2. b) The probability that at least one of the two buses arrives late.

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

1. a) Write down the sample space of this compound 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?

A lottery drum contains 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 the data from the last $2000$ chests opened by players, the following results have been obtained:

• Common item: $1240$ times.

• Rare item: $680$ times.

• Legendary item: $80$ times.

1. a) Calculate the empirical probability (relative frequency) of getting a legendary item.
2. b) If the developer states that the theoretical probability of getting a rare or legendary item is $40\%$, do you think the algorithm is working correctly according to the Law of Large Numbers? Justify 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 the events are independent).

Two esports teams, the "Pixels" and the "Lag-Killers", are playing the final of a best-of-three tournament (the first to get two wins is the winner). From previous statistics, we know that in each individual match, the "Pixels" have a $P(P) = 0,6$ probability of winning, while the "Lag-Killers" have $P(L) = 0,4$. There is no possibility of a tie.

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" make a comeback and win the tournament.

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

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

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

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

1. a) Calculate the probability that a parcel 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 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 the 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.

A study has been carried out on the use of social media in a middle school of $300$ students. The results are as follows:

• $180$ students use Instagram.

• $130$ students use TikTok.

• $40$ students do not use either of the two networks.

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) Given that the student uses TikTok, what is the probability that they also use Instagram?

A multiple-choice exam consists of $4$ questions with $3$ answer 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 questions right?
2. b) What is the probability that they get exactly two questions right?
3. c) If they need to get at least $3$ questions right to pass, what is the probability of them 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 the other 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.