Probability (DP IB Analysis & Approaches (AA): HL): Exam Questions

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

Using an appropriate representation, describe the sample space of possible outcomes when the spinner is spun and the dice is rolled at the same time.

When the game is played, the spinner is spun and the dice 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.

A survey was carried out of residents of a particular town, to find out what their preferred activity was when coronavirus lockdown restrictions were in place.

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 during lockdown

(ii) was less than 55 years old and did not prefer daydreaming during lockdown.

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 during lockdown

(ii) were between 31 and 70 years old and did not prefer exercising during lockdown.

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.

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.

264 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 and B are events such that  , , and   .

By drawing a Venn diagram to illustrate these probabilities, find

 (i)    

 (ii)   

(iii)   

 and  are independent events, such that    and  .   is another event, such that  and  are mutually exclusive and  .

Given that  ,  find

(i)

(ii)

(iii)

(iv)

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.

In a game of Unicorns Versus Zombies your unicorn is attempting to use the magic of its horn do 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.

For two events,  and , the conditional probability of  occurring given that  has occurred is given by

If in addition events  and  are independent, then it will also be true that

Given that events  and  are independent, use the results above to show that

(i)    

(ii)    .

Sciura is a scientist who studies grey squirrel populations in Canada.  Some grey squirrels actually have black fur, and Sciura believes that the probability of a grey squirrel having black fur is independent of whether the squirrel is male or female.

Because of coronavirus lockdown restrictions, Sciura and her team have been unable to get into the field to conduct research to test her belief.  However Sciura has managed to locate scientific publications in which the results of past studies of grey squirrel populations in Canada are reported.

In one study, a random sample of 98 291 grey squirrels was looked at and the authors noted that 297 of the squirrels had black fur.

In another study, a random sample of 13 583 female grey squirrels was looked at and the authors noted that 41 of the squirrels had black fur.

Clearly stating any assumptions you make, state with a justification whether or not the results of these two studies support Sciura’s belief.

100 people are queuing to buy tickets for a live concert by a particular band. When asked, 90% said they had seen the band play live on a previous occasion.  Of those 90%, 70% said they had been queueing longer than an hour.  Of those who had not seen the band play live before, 80% said they had been queueing an hour at maximum.

Draw a tree diagram to represent this situation.

Write down the probability that a randomly selected person from the queue has been

(i) queueing longer than an hour, given that they had previously seen the band play live,

(ii)  queueing longer than an hour, given that they had not previously seen the band play live.

Find the probability that a randomly selected person from the queue has seen the band play live previously, given that they have been queueing more than an hour.

In a computer game, a player has to capture a variety of creatures and can offer different types of fruit to the creatures in order to capture them. The decision a player makes regarding which fruit to feed a creature affects how likely they are to be captured. Players have a choice of three options; strawberries, raspberries or blueberries.

A particular player always feeds a creature fruit before attempting to capture it. They randomly select a strawberry on 30% of their attempts at capturing a creature, 40% of such attempts result in the creature being successfully captured. The player selects a raspberry 50% of the time with a 50% success rate whilst blueberries yield a 30% success rate.

(i) Given that the player successfully captures a creature, find the probability they fed it a raspberry.

(ii) Given that the player fails to capture a creature, find the probability they fed it a blueberry. 

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

Using an appropriate representation, describe the sample space of possible outcomes when the spinner is spun and the dice is rolled at the same time.

When the game is played, the spinner is spun and the dice 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 dice, then the score is the sum of the two numbers;

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

• if the numbers on the spinner and the dice 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.

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 while they were stuck at home during coronavirus lockdown. 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 during lockdown

(ii) were less than 55 years old and had chosen Hittite or Mycenaean Greek as their main language during lockdown.

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 Aaplunge.

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 and.

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 events such that  ,  and 

Find:

(I)

(ii)

(iii)

A, B  and C are three events such that ,  and events B and C  are independent.  Additionally, and .

Given that  and , find:

(i)

(ii)

(iii)

(iv)

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.

Explain why the events “both tokens drawn were the same colour” and “the coin landed on tails” are not independent events.

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

For two events,  and , the conditional probability of  occurring given that  has occurred is given by

If in addition events  and  are independent, then it will also be true that

Given that events  and  are independent, use the results above to show that

(i)    

(ii)    

Events  and  are independent.  Given that  is twice as large as , and also that  ,  find  and .

A pharmaceutical company is testing out its “regular” antiseptic cream against a new version it has developed named “Lots-on”.

In trials, 40% of volunteers were given the regular cream whilst the rest were given “Lots-on”.

Of those testing the regular cream on a cut on their finger, 15% reported the cut healed within 2 days; 35% reported the cut took 2 - 4 days to heal and the rest reported it took longer than 4 days.

Of those testing “Lots-on”, 30% reported the cut healed within 2 days; 45% reported the cut took 2 - 4 days to heal and the rest reported it took longer than 4 days. 

Draw a tree diagram to represent this information.

(i) Find the probability that a cut taking between 2 and 4 days to heal had “Lots-on” cream applied to it.

(ii) Find the probability that “Lots-on” cream was used for a cut taking longer than 4 days to heal.

Three independent events, and are such that and

It is also known that for event and

Find the value of a, given that Fully justify your solution.