Probability (DP IB Maths: AA SL)

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

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:

		Preferred activity
		Daydreaming	Staring at Phone	Exercising	Playing Chess	Other
Age	13-17	11	37	33	1	2
	18-30	2	45	40	1	1
	31-54	33	8	31	21	8
	55-70	31	35	30	11	10
	70	34	17	38	13	7

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

The town has a total population of .

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

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

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

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

 represents the event that the member has dressed as Aquaman.

 represents the event that the member has dressed as Batman.

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

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

determine the values of

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.

 and  are events such that  ,  ,  and  .

By drawing a Venn diagram to illustrate these probabilities, find

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

Given that  ,  find

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

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

the flies are turned into glitter pixies

the flies are dispelled

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

.

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.

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

exactly 7

10 or more

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:

		Main language
		Akkadian	Hittite	Mycenaen Greek	Middle Persian	Old Church Slavonic
Age	13-17	5	3	13	2	4
	18-30	9	11	10	15	13
	31-54	16	15	12		12
	55-70	10	10	11	5	5
	70		5	9	4	3

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

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 , the Barnstormer rollercoaster ( ), and the Really Scary Carousel ( ).

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:

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 ( ), spaghetti Bolognese ( ), and linguine carbonara ( ).

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

had sampled exactly two of the three dishes

had sampled at least one of the three dishes but not all three of them.

 and  are events such that  ,  ,  and 

Find:

.

,  and  are three events such that  ,  ,  and events  and  are independent.  Additionally,    and  .

Given that  ,    and   , find:

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

the zombie is destroyed

your character dies

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

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