Probability (DP IB Applications & Interpretation (AI): SL): Exam Questions

The Venn diagram below shows the number of members of an amateur Elizabethan dramatic society who have been involved with productions of the following three plays by Ben Jonson: The Alchemist (A), Bartholomew Fayre (B) and Chloridia (C).

There are 150 members of the society in total.

 Given that the probability of a member having been involved with a production of Chloridia is  ,$\frac{8}{25}$,

determine the values of

(i) x

(ii) y.

Determine the probability that a member of the society

(i) has been involved with a production of at least one of the three plays

(ii) has been involved with a production of exactly one of the three plays.

The following Venn diagram shows the number of adults in a poll who said they enjoy watching action films (A), Bollywood musicals (B), and crime thrillers (C).

100 adults were polled in total.

One of the adults who was polled is selected at random. Given that the adult chosen enjoys watching at least one of those three genres of film, find the probability that the adult enjoys watching:

(i) Bollywood musicals

(ii) only one of the three genres of film

(iii) exactly two of the three genres of film.

One of the 100 adults is selected at random. Find the following probabilities:

(i) $\mathrm{P}(A\cap C)$

(ii) $\mathrm{P}(A\cup C)$

(iii) $\mathrm{P}\left(C\right|B)$

(iv) $\mathrm{P}(B\text{'})$

On any given day the probability that Radagast has a lichen smoothie with his lunch is 0.4, and the probability that he has a wild mushroom wrap is 0.8. Given that the probability of him having both those items is 0.35, find the probability that Radagast has:

(i) a wild mushroom wrap but not a lichen smoothie

(ii) neither a wild mushroom wrap nor a lichen smoothie.

$A$ and $B$ are two events such that $\mathrm{P}\left(A\right)=0.35$, $\mathrm{P}\left(B\right)=0.25$ and $\mathrm{P}(A\cup B)=0.6$.

Determine whether $A$ and $B$ are mutually exclusive. Justify your answer.

$C$ and $D$ are two events such that $\mathrm{P}\left(C\right)=0.2$, $\mathrm{P}\left(D\right)=0.4$ and $\mathrm{P}(C\cap D)=0.18$.

Determine whether $C$ and $D$ are independent. Justify your answer.

A bag contains 13 yellow tokens and 7 green tokens. Two tokens are drawn from the bag without replacement.

Draw a tree diagram to represent this experiment.

Find the probability that the two tokens drawn are the same colour.

Ichabod is a keen chess player who plays one game of chess online every night before going to bed.  In any one of those games, the probabilities of Ichabod winning, drawing, or losing are 0.4, 0.27 and 0.33 respectively. Following each game, the probabilities of Ichabod sleeping well after winning, drawing or losing are 0.7, 0.9 and 0.2 respectively.

Draw a tree diagram to represent this information.

Find the probability that on a randomly chosen night

(i) Ichabod loses his chess game and sleeps well

(ii) Ichabod sleeps well.

Given that Ichabod sleeps well, find the probability that his chess game did not end in a draw.

$A$ and $B$ are events such that $\mathrm{P}\left(A\right)=0.24$, $\mathrm{P}\left(B\right)=0.73$ and $\mathrm{P}(A\cup B)=0.84$.

Find

(i) $\mathrm{P}(A\text{'}\cup B)$

(ii) $\mathrm{P}(A\cap B\text{'})$

(iii) $\mathrm{P}\left(\right(A\cap B)\text{'})$

The Venn diagram illustrates the probabilities of members of a costumed performers’ union having dressed as one or another superhero during a performance.

$A$ represents the event that the member has dressed as Aquaman.

$B$ represents the event that the member has dressed as Batman.

$C$ represents the event that the member has dressed as Captain Marvel.

Given that the probability of a member having dressed as Captain Marvel is $0.44$,

determine the values of

(i) $x$

(ii) $y$

256 of the union’s members have dressed as exactly two of the three superheroes.

Use this information to determine the total number of members of the union.

A bag contains 12 red marbles, 7 green marbles and 1 black marble. Two marbles are drawn from the bag without replacement.

Draw a tree diagram to illustrate the process described above, showing clearly the probabilities on each branch.

Find the probability that

 (i)     the two marbles drawn are not both the same colour

 (ii)    both marbles are green, given that both marbles drawn are the same colour.

In the context of the question, give an example of two mutually exclusive events. Be sure to justify that they are mutually exclusive.

A game is played using a fair spinner with four sectors numbered 1 to 4, as well as a fair die with its six faces numbered 1 to 6.

Draw a sample space diagram to show all the possible outcomes when the spinner is spun and the die is rolled at the same time.

When the game is played, the spinner is spun and the die is rolled at the same time, and the player’s score is defined to be the (positive) difference between the two results.

Find the probability of the score in the game being

(i) exactly 0

(ii) 3 or more

(iii) a prime number

The game is played 150 times.

Find the expected number of times that a prime number score will occur.

The Venn diagram displays information about the number of students taking each of three languages: Mandarin Chinese ($C$), German ($G$) and Latin ($L$).

There are fifty students in total.

Determine the number of students who take only Latin.

A student is randomly chosen from the group.

Find the probability that

(i) the student studies German or Latin

(ii) the student studies neither Mandarin Chinese nor Latin

(iii) the student studies Mandarin Chinese, given that they study German

(iv) the student studies Latin, given that they study Mandarin Chinese

(v) the student studies Latin, given that they do not study German.

A survey was carried out of residents of a particular town, to find out what their preferred activity was when stuck at home. Five hundred residents were surveyed, and the results are shown in the table below:

One of the surveyed residents is selected at random. Given that the resident did not give a response of ‘Other’ to the survey, find the probability that this resident

(i) preferred playing chess when stuck at home

(ii) was less than 55 years old and did not prefer daydreaming when stuck at home.

The town has a total population of $23681$.

Assuming that the survey figures are representative of the town as a whole, estimate the number of residents of the town who

(i) preferred daydreaming, exercising, staring at their phone or playing chess when stuck at home

(ii) were between 31 and 70 years old and did not prefer exercising when stuck at home.

A game is played using a fair spinner with four sectors numbered 1 to 4, as well as a fair eight-sided die with its faces numbered 1 to 8.

Draw a sample space diagram to show all the possible outcomes when the spinner is spun and the die is rolled at the same time.

When the game is played, the spinner is spun and the die is rolled at the same time, and the player’s score is determined as follows:

• if the number on the spinner is higher than the number on the die, then the score is the sum of the two numbers;

• if the number on the spinner is lower than the number on the die, then the score is the (positive) difference of the two numbers;

• if the numbers on the spinner and the die are equal, then the score is the product of the two numbers.

Find the probability of the score in the game being

(i) exactly 7

(ii) 10 or more

(iii) a triangular number (1, 3, 6, 10, 15, 21, …).

The game is played 300 times. Find the expected number of times that a triangular number score will occur.

$A$ and $B$ are events such that $\mathrm{P}\left(A\right)=0.58$, $\mathrm{P}\left(B\right)=0.71$ and $\mathrm{P}\left(\right(A\cup B)\text{'})=0.27$.

Find:

(i) $\mathrm{P}(A\text{'}\cup B)$

(ii) $\mathrm{P}(A\text{'}\cap B\text{'})$

(iii) $\mathrm{P}(A\text{'}\cup B\text{'})$

The Venn diagram below shows the probabilities of attendees at a charity pasta dinner having sampled one of the three pasta dishes on offer: alphabetty spaghetti ($A$), spaghetti Bolognese ($B$), and linguine carbonara ($C$).

Given that half the attendees sampled the linguine carbonara, and that 38% of the attendees sampled at least two of the three dishes, determine the values of $x$, $y$ and $z$.

An attendee from the dinner is chosen at random.

Determine the probability that the attendee

(i) had sampled exactly two of the three dishes

(ii) had sampled at least one of the three dishes but not all three of them

(iii) had not sampled any of the pasta dishes, given that they had sampled less than two of them.

$A$ and $B$ are independent events, such that $\mathrm{P}\left(A\right)=0.25$ and $\mathrm{P}\left(B\right)=0.52$. $C$ is another event, such that $B$ and $C$ are mutually exclusive and $\mathrm{P}(A\cap C)=0.09$.

Given that $\mathrm{P}(A\cup B\cup C)=0.95$, find

(i) $\mathrm{P}(A\cap B)$

(ii) $\mathrm{P}\left(C\right)$

(iii) $\mathrm{P}(A\text{'}\cap B\text{'})$

(iv) $\mathrm{P}\left(A\right|C\text{'})$

In a game of Unicorns Versus Zombies your unicorn is attempting to use the magic of its horn to dispel a cloud of zombie apocalypse flies.

On the first attempt, the probability of the magic working is 0.7.  If the magic works, then there is a probability of 0.2 that the flies will be turned into glitter pixies and join your rainbow army, otherwise the flies will simply be dispelled.

If the magic does not work the first time you may try again, although the probability of your magic working the second time is only 0.6.

Similarly, if your magic does not work the second time you may try a third time, but on the third attempt the probability of your magic working is reduced to 0.5.

If your magic works on the second or third attempts the probabilities of dispelling the flies or turning them into glitter pixies are the same as for the magic working on the first attempt.  If your magic does not work on the third attempt, however, then your unicorn is turned into an evil zombiecorn and joins the zombie horde.

In all cases, the game ends when either the flies are turned into glitter pixies, or the flies are dispelled, or your unicorn is turned into a zombiecorn.

Draw a tree diagram to illustrate the above question, showing clearly the probabilities on each branch.

Find the probability that

(i) the flies are turned into glitter pixies

(ii) the flies are dispelled

(iii) your unicorn is turned into a zombiecorn.

Explain why the events “the flies are turned into glitter pixies” and “the magic worked on the second attempt” are not independent events.

The game Undead Redemption is played using three fair dice: a four-sided die with the sides numbered 1 to 4, a six-sided die with the sides numbered 1 to 6, and an eight-sided die with the sides numbered 1 to 8.

In the game your character is battling a zombie. The battle can last between one and three rounds, and it is resolved as follows:

• In the first round, you and the zombie each roll the four-sided die. If your roll is greater than or equal to the zombie’s roll then the zombie is destroyed and the battle is over. Otherwise your character is wounded and the battle goes on to the second round.

• In the second round, you roll the four-sided die and the zombie rolls the six-sided die. If your roll is greater than or equal to the zombie’s roll then the zombie is destroyed and the battle is over. Otherwise your character is wounded again and the battle goes on to the third round.

• In the third round, you roll the four-sided die and the zombie rolls the eight-sided die. If your roll is greater than or equal to the zombie’s roll then the zombie is destroyed and the battle is over. Otherwise your character is wounded for the third time and dies.

Draw a tree diagram to represent this information.

Find the probability that

(i) the zombie is destroyed

(ii) your character dies

(iii) the zombie is destroyed, given that your character is wounded one or more times

In the context of the question, give an example of two mutually exclusive events. Be sure to justify that they are mutually exclusive.

A bag contains 10 black tokens and 6 white tokens.  A token is drawn from the bag and its colour recorded, and then a fair coin is flipped.  If the coin lands on heads then a second token is drawn from the bag without replacing the first token.  If the coin lands on tails then the first token is replaced in the bag before a second token is drawn.

Draw a tree diagram to represent this experiment.

Find the probability that the second token drawn is white.

Determine whether the events “both tokens drawn were the same colour” and “the coin landed on tails” are independent. Justify your answer.

Leofranc is the membership secretary of an ancient languages enthusiasts’ society. He conducted a survey to discover what the main language was that society members had chosen to study most recently. Some of the results of this survey are contained in the following table:

Unfortunately Leofranc spilled gallic acid on the survey results, so the numbers that belong in the two empty boxes on the table can no longer be read. Leofranc remembers, however, that the number of people who had chosen Middle Persian as their main language was only half the number of those who had chosen Akkadian. Also, the events “had chosen Hittite as their main language” and “was between 18 and 30 years old” were independent.

Use the above information to complete the table.

The society has a total of 1138 members (not all of whom responded to the survey). Assuming that the survey figures are representative of the society as a whole, estimate the number of members of the society who

(i) had not chosen Akkadian as their main language to study most recently

(ii) were less than 55 years old and had chosen Hittite or Mycenaean Greek as their main language to study most recently.

120 students went on a school trip to the Thormton Manor theme park. A statistics student has begun filling in the following Venn diagram, showing the numbers of students who went on none, one or more of the park’s three most terrifying rides: the Aquaplunge water slide ($A$), the Barnstormer rollercoaster ($B$), and the Really Scary Carousel ($C$).

A student is randomly chosen from the group that went to the theme park.

Given that ‘went on Aquaplunge’ and ‘went on the Really Scary Carousel’ were mutually exclusive events, while ‘went on Aquaplunge’ and ‘went on the Barnstormer’ were independent events, find the probability that the student:

(i) went on the Really Scary Carousel

(ii) did not go on exactly two of the rides

(iii) went on Aquaplunge, given that they went on the Barnstormer

(iv) went on the Barnstormer, given that they went on less than two of the rides

(v) went on the Really Scary Carousel, given that they did not go on Aquaplunge.

$A$, $B$ and $C$ are three events such that $\mathrm{P}\left(A\right)=0.2$, $\mathrm{P}\left(B\right)<0.5$, and events $B$ and $C$ are independent. Additionally, $\mathrm{P}(A\cap B)=0.01$ and $\mathrm{P}(A\cap C)=0.14$.

Given that $\mathrm{P}(B\cap C\text{'})=0.03$, $\mathrm{P}(B\text{'}\cap C)=0.38$ and $\mathrm{P}(A\cap B\cap C)=\mathrm{P}(A\text{'}\cap B\cap C)$, find:

(i) $\mathrm{P}(B\cap C)$

(ii) $\mathrm{P}(C\text{'})$

(iii) $\mathrm{P}(A\text{'}|C)$

(iv) $\mathrm{P}(A\cup B\cup C|B\text{'})$