Probability of equally likely outcomes

A fair die is rolled. Find the probability of each event:

a) Rolling a $3$.

b) Rolling an even number.

c) Rolling a number greater than $4$.

d) Not rolling a $6$.

A card is drawn from a Spanish deck of $40$ cards. Find the probability that it is:

a) Coins (Oros).

b) An ace.

c) A picture card (jack, knight, or king).

d) The king of cups.

In an urn there are $4$ red balls and $6$ green balls. How many blue balls must be added so that the probability of drawing a red ball is $\frac{1}{4}$? Set up the equation and solve it.

a) If the probability of drawing diamonds in a standard deck of cards is $\frac{1}{4}$, what is the probability of not drawing diamonds?

b) If $P(A) = 0.35$, what is the probability of the complementary event of $A$?

c) An urn contains only red, blue and green balls. If the probability of drawing a red ball is $\frac{2}{7}$ and that of drawing a blue ball is $\frac{3}{7}$, what is the probability of drawing a green ball?

In a bag there are $5$ red balls, $3$ blue balls and $2$ white balls. A ball is drawn at random. Calculate the probability that:

a) It is not white.

b) It is white or blue.

c) It is yellow.

d) It is not yellow.

In a cooperative board game, to activate a protective spell, $3$ fair six-sided cubic dice are rolled at the same time. The condition for the spell to work is to roll at least one $6$ among the three dice. Two students discuss how to calculate the probability of success:

• Aoife suggests: "It is much faster to first calculate the probability of not rolling any $6$ on any of the dice and then apply the complementary event property: $P(\text{win}) = 1 - P(\text{no } 6)$"

• Seán suggests: "It is better to calculate the favorable outcomes by adding three different situations: rolling exactly one $6$, rolling exactly two $6$s, or rolling three $6$s".

a) Determine the total number of possible outcomes when rolling the three six-sided dice simultaneously.

b) Work through Aoife's method step by step: calculate how many outcomes do not contain any $6$, obtain the probability of not winning, and deduce the final probability of winning as an irreducible fraction and an approximate percentage.

c) Work through Seán's method: count in a reasoned and organized way how many possible outcomes correspond to rolling exactly one $6$, how many to exactly two $6$s, and how many to three $6$s. Verify that the total sum of favorable outcomes matches Aoife's result exactly.

d) Justify with reasons which of the two methods you would choose if the game rule changed and $5$ dice were rolled instead of $3$, evaluating the difficulty of counting in each approach.

A coin is tossed and a die is rolled. Find the probability of getting:

a) Heads and a $6$.

b) Tails and an odd number.

c) Heads and a number less than $5$.

Two dice are rolled and the sum of the scores is recorded. Use a sample space table to calculate:

a) $P(\text{sum } 7)$

b) $P(\text{sum greater than } 9)$

c) $P(\text{even sum})$

d) $P(\text{sum other than } 7)$

Three coins are tossed. Using a tree diagram, find the probability of getting:

a) Exactly two heads.

b) At least one tail.

c) Three identical outcomes.

d) Exactly one head.

The students at a school who do extracurricular sports are distributed as follows:

A student is chosen at random. Calculate the probability that:

a) They are in Primary.

b) They do swimming.

c) They are in Primary and play tennis.

d) Given that they are in Junior Cycle, they play tennis.

In a class of $30$ students there are $18$ girls, of whom $12$ have passed all subjects. In total, $10$ students in the class have failed at least one subject.

a) Construct a two-way table (girls/boys versus passed everything/failed at least one subject).

b) If a person is chosen at random, what is the probability that they are a boy and have passed everything?

c) Given that the chosen person has failed at least one subject, what is the probability that she is a girl?

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

Two cards are drawn from a deck of $40$ cards (Spanish-style, with four suits), without replacing the first card in the deck.

a) What is the probability that the first card is coins?

b) If the first card was coins, what is the probability that the second card is also coins?

c) What is the probability that both cards are coins?

d) And what is the probability that neither card is coins?

The spinner shown in the figure has three sectors with the indicated angles.

Find the probability that the arrow stops on:

a) The red sector.

b) The green sector.

c) A sector that is not yellow.

The spinner shown in the figure has four sectors with the indicated angles.

Find the probability that the arrow stops on:

a) The red sector.

b) The yellow sector.

c) The blue sector or the green sector.

d) A sector that is not red.

A youth association designs a $360^\circ$ circular spinner to organize a charity raffle and raise funds for a social cause. The spinner has four sectors: Gold, Silver, Bronze, and White (no prize). The guidelines agreed upon for its design are:

• The probability of landing on the "Gold" sector must be exactly half that of landing on the "Silver" sector.

• The "Bronze" sector must cover a central angle of $90^\circ$.

• The probability of not winning a prize ("White" sector) must be exactly $0.4$.

• Each spin will cost participants $1$ euro, and a total of $600$ spins are expected to take place throughout the day.

a) Deduce the central angle in degrees ($^\circ$) and the theoretical probability corresponding to each of the four sectors of the spinner.

b) The established prizes are: the "Gold" sector awards a $5$-euro voucher, the "Silver" sector awards a $2$-euro voucher, the "Bronze" gives a commemorative souvenir that cost the association $0.50$ euros, and the "White" gives nothing. Calculate the total expected expenditure on prizes if the $600$ planned spins are carried out.

c) Compare the expected revenue from selling the $600$ spins with the estimated prize expenditure: will the association manage to raise profits with this design, or will it incur losses? Justify your numerical answer and propose a concrete change in the distribution of the sectors to correct the situation while keeping all four types of prizes.