Basic Probability (CIE AS Maths: Probability & Statistics 1)

and  are two events such that  ,   and  .  State, with a reason, whether and   are mutually exclusive.

and are two events such that  ,    and  . State, with a reason, whether and   are independent.

The lengths, in cm, of 120 adult platypuses are recorded in the following table:

Length, l (cm)	Frequency (female)	Frequency (male)
39 ≤  < 42	14	0
42 ≤  < 45	29	0
45 ≤  < 48	12	7
48 ≤  < 51	6	21
51 ≤  < 54	3	19
54 ≤  < 57	1	5
57 ≤  < 60	0	2
60 ≤ < 63	0	1

One platypus is chosen at random.  Find the probability that the platypus is:

Two fair spinners each have three sectors numbered 1 to 3.  The two spinners are spun together and then the product of the numbers indicated on each spinner is recorded.

Find the probability of the product indicated by the spinners being

A fair four-sided die and a fair six-sided die are rolled and the sum of the numbers on their uppermost faces is recorded. 

Complete the sample space below showing all possible outcomes.

An event is ‘the sum on the two dice is more than 7’ and an event  is ‘scoring an odd number on one die and an even number on the other’. 

Find: 

Determine whether the events  and are independent or not, justify your answer.

Two words are chosen at random from the word IMPOSSIBLE without replacement.

The event is defined as ‘choosing two of the same letters’ and the event   is defined as 'the first letter taken is a letter S’. 

Find:

Determine whether the events  and  are mutually exclusive, justify your answer.

Angelo meets with his maths tutor, Miss Perry, once a week.  The probability that Miss Perry changes the time of the meeting is 0.2.  The probability that she is late is 0.15.  Otherwise she is on time.  For the next two weeks that they meet, find the probability that Angelo’s maths tutor:

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.

In a game of Galactic Unicorns your Monocerian-class space frigate is attacking an evil Sargonian robot ship.  Your attack will either hit or miss the robot ship, the probability of hitting the ship is 0.7.  If you hit the robot ship then there is a probability of 0.8 that the ship will be destroyed, otherwise it will manage to escape.  If you miss the robot ship then there is a probability of 0.2 that it will manage to escape, otherwise it will surrender because it has witnessed the immense power of your rainbow lasers. 

Draw a tree diagram to represent this information.

Find the probability that the robot ship

Show that the events ‘you hit the robot ship’ and ‘the robot ship manages to escape’ are independent.

A spinner with five equal sections labelled 1, 2, 3, 4 and 5 is spun once, the number the spinner lands on is recorded and then it is spun again. The event   is ‘The product of the two numbers the spinner lands on is even’ and the event is ‘The first number is greater than the second number’. 

Construct a sample space diagram to display the product of the two numbers when the spinner is spun twice.

Find:

Give an example to show that the events  and are not mutually exclusive.

The histogram below shows the distribution of masses, in grams, of 80 newly hatched ducklings:

Find the probability that a duckling chosen at random has a mass less than

Estimate the probability that a duckling chosen at random has a mass greater than

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.  The spinner is spun and the dice is rolled, and the score in the game is defined as the (positive) difference between the two results.

Find the probability of the score in the game being

On any given day the probability that Tomás works on his poetry is 0.55, and the probability that he stops in at his local pub is 0.75.  Given that he always does at least one of those two things, find the probability that on a given day Tomás:

are four events such that  and  

Given that are mutually exclusive, find

Given that are independent, find .

Praewa plays sports every day after school.  She is four times more likely to choose football than tennis and three times more likely to choose basketball than tennis. She does not play any other sport. 

Show that the probability of Praewa choosing tennis is .

Find the probability that Praewa chooses three different sports on the next three days in a row.

On a five day week chosen at random, find the probability that Praewa chooses football every day.

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 represent this experiment.

Find the probability that the two marbles drawn are not both the same colour.

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 zombie corn 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 zombie corn.

Draw a tree diagram to represent this information.

Find the probability that

Give an example of two events in the game that are mutually exclusive.

A six -sided spinner with equal segments labelled 1, 1, 1, 2, 3, 3 is spun and a fair coin is tossed.  If the coin lands on heads the number recorded is one more than the number on the spinner. If the coin lands on tails the number on the spinner is doubled and recorded.  The event A is ‘the number recorded is a prime number’ and the event B is ‘the number recorded is not one of the original numbers on the spinner’. 

Show that the events A and B are mutually exclusive, justify your answer.

One of the segments on the spinner showing the number 1 is changed to a number 4 and the experiment is repeated using the same rules as before.  

Find:

After the change that has been made to the spinner, determine if the events A and B are still mutually exclusive. Justify your answer.

The histogram below shows the distribution of masses, in kg, of 100 newborn calves:

Find the probability that a calf chosen at random has a mass greater than

Estimate the probability that a calf chosen at random has a mass less than .

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. The spinner is spun and the dice is rolled, and the score in the game 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

On any given day the probability that Björn has an avocado with his lunch is 0.39, while the probabilities of him having a bacon butty or a piece of carrot cake are 0.32 and 0.44 respectively.  He never has all three items on the same day, but the probability that he doesn’t have at least one of them is only 0.09.  Given that the probability of him having exactly two of the three items is the same regardless of which two items those are, find the probability that on a given day Björn has:

are three events such that    and

Given that and are mutually exclusive, and thatand are independent, find and

Isabel has two boxes of chocolates.  Box contains 4 caramels and 6 toffees, and box contains 5 caramels and  toffees.  Isabel chooses a chocolate at random from box A and moves it to box B.  Her friend then takes a chocolate at random from box B. 

Show that the probability Isabel’s friend takes a caramel is given by . 

Given that the probability that Isabel’s friend takes a toffee is , find the value of .

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.

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

your character dies