2.3 Further Probability

Some attendees at a pizza fandom convention are surveyed regarding their opinions about anchovies and bananas as pizza toppings.  144 of them say they do not like anchovies. 320 of them say they do not like bananas.  28 of them say they like bananas but not anchovies. Only 12 of them like both toppings. 

Let  and  be the events ‘likes anchovies as a pizza topping’ and ‘likes bananas as a pizza topping’ respectively. 

Draw a two-way table to show this information.

One of the attendees is chosen at random. Find:

and are two events with   and  ,  where .  Given that and  are independent, find the following probabilities in terms of x and y:

A group of 18- to 25-year-olds, and a group of people over 65 years old, were asked whether they would prefer to holiday in Ibiza or Skegness. The following two-way table shows part of the results of the survey:

Farah is playing a game with a coin.  She tosses the coin three times and awards herself points using the following rules. If the coin lands heads up she gets one point and if the coin lands tails up she gets two points.  However, if the coin lands the same way it did on the previous coin toss, she awards herself two points for heads up and four points for tails up.  The final score is the sum of the points awarded from the three tosses of the coin.

The event  is ‘Farah’s final score is 5 points’ and event is ‘the coin lands on heads at least two times’.  Determine whether the events  and  are independent.  Justify your answer.

Olivia owns a pottery business where she runs workshops on Saturdays and sells pots during the week.  If she has run a workshop on the previous Saturday, she will open the shop for a short week, Monday to Thursday, with a probability of 0.6.  If she did not run a workshop, she will open the shop for a short week, Monday to Thursday, with a probability of 0.2.  Otherwise, she will open the shop for a long week, Monday to Friday.  The probability that Olivia opens the shop for a short week, Monday to Thursday, on any given week is 0.46.  

Find the probability that Olivia runs a workshop on any given week.

Given that Olivia opened the shop for a short week, Monday to Thursday, on a particular week, find the probability that she ran a workshop the previous Saturday.

Shalit has seven identical red socks and  identical blue socks in his drawer.  He takes two socks out of his drawer at random and puts them on. 

Find expressions for  on the tree diagram above.

The probability that Shalit took two red socks, given that he took two socks of the same colour is . Find the value of .

A bag contains 12 orange marbles, 8 purple marbles and 5 red marbles.  A marble is taken from the bag and its colour is recorded, but it is not replaced in the bag.  A second marble is then taken from the bag and its colour is recorded.

Draw a tree diagram to represent this information.

Find the probability that:

Rosco is a somewhat inept rural county sheriff who frequently finds himself involved in car chases with well-meaning local entrepreneurs.  During any given car chase, Rosco inevitably runs into one of three obstacles – a damaged bridge (with probability 0.47), an oil slick (with probability 0.32), or a pigpen at the end of a dead-end road.

If he encounters a damaged bridge there is a 25% chance that he will make it across safely; otherwise he lands in the river and ends up covered in mud.  If he encounters an oil slick there is a 40% chance that his car will spin around and he will end up continuing his hot pursuit in the wrong direction; otherwise he goes off the road into a farm pond and ends up covered in mud.  If he encounters a pigpen at the end of a dead-end road there is a 15% chance he will stop his car in time; otherwise he drives into the muddy end of the pigpen while the pigs sit at the other end laughing.  If he drives into the muddy end of a pigpen there is a 20% chance he will only end up covered in mud; otherwise he ends up covered in mud and other things that are found in pigpens.

Draw a tree diagram to represent this information.

Find the probability that in the course of a randomly chosen car chase

Given that Rosco ends up covered in mud in the course of a randomly chosen car chase, find the probability that he didn’t encounter an oil slick. Give your answer as an exact value.

In the course of a particular day Rosco finds himself engaged in three separate car chases with well-meaning local entrepreneurs.  The car chases may be considered to be independent events.

Determine the probability that on that day Rosco will not end up covered in other things that are found in pigpens.

240 students are surveyed regarding their belief in supernatural creatures. 144 say they believe in unicorns . 75 say they believe in vampires . Of those who believe in vampires, 27 also believe in unicorns. 

Draw a two-way table to show this information.

One student is chosen at random. Find:

and are two events with  and    Given that and  are independent, write down

A group of middle and senior school students were asked whether they preferred vinegar or ketchup as a topping on their chips. The following two-way table shows the results of the survey:

Katin is collecting tadpoles for her biology project.  In the pond there are five female tadpoles and six male tadpoles, however they all look the same. She takes two tadpoles at random from the pond, without replacement. 

Draw a tree diagram to represent this information.

Find the probability that Katin’s two tadpoles are

A fair hexagonal spinner with seven equal sections numbered 0, 1, 1, 3, 3, 3 and 3 is spun twice and the sum of the two numbers the spinner lands on is recorded.

Draw a sample space to represent the outcomes of this experiment.

S is the event ‘the sum of the two numbers is a square number’ and E is the event ‘the sum of the two numbers is a positive even number’. Use your sample space to find:

Use your answers to part (b) to show that

A standard pack of 52 playing cards is comprised of thirteen different cards for each of the four suits: hearts, diamonds, spades and clubs.  Each suit is made up of nine different number cards (two to ten) and four court cards (jack, queen, king and ace).  A full pack of cards is shuffled and one card is selected at random.  Let  be the event ‘the card chosen is a heart’.  Let  be the event ‘the card chosen is a court card (jack, queen, king or ace)’. 

Find

Use your answers to part (a) to show that the events C and H are independent events. 

A bag contains 15 blue tokens and 27 yellow tokens.  A token is taken from the bag and its colour is recorded, but it is not replaced in the bag.  A second token is then taken from the bag and its colour is recorded.

 Draw a tree diagram to represent this information.

Find the probability that:

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

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

A spinner has five sections numbered . The probability the spinner lands on each section,  is given by the model:

Find the value of .

Draw a two-way table of probabilities for the events ‘the spinner lands on an odd number’ and ‘the spinner lands on a prime number’.

Find:

An online chess competition is being played where participants can either win, draw or lose each game.  If the participant wins, they can play again straight away.  If the participant loses, they are out of the competition.  If there is a draw, the participants will undertake a sudden death match until there is a winner.  

The probability that Jamie will win his first match outright is 0.7; the probability that he will lose it outright is 0.1.  If Jamie wins his first match outright the probabilities that he wins or loses his second match outright remain the same.
However, if he plays sudden death in his first match and wins, the probability he will win his second match outright is 0.55 and the probability he will lose it outright is 0.3.
The probability Jamie wins at sudden death is always the same. 

Given that the probability Jamie wins his first two matches is 0.6508, find the probability that Jamie wins any match at sudden death.

Find the probability that Jamie won both of his matches outright, given that he won his first two matches.

The cost of Lucy’s three favourite snacks at the two nearest movie theatres to her house - The Amenic and The Seivom - are summarised in the table below.  

When Lucy goes to the movies she always chooses one of these two theatres and always has one, and only one, of these three snacks.
If Lucy chooses The Amenic, she chooses popcorn with a probability of 0.2 and ice cream with a probability of 0.5.
If Lucy chooses The Seivom she will have nachos with a probability of 0.4 and ice cream with a probability of 0.2.

Given that the probability Lucy spends exactly $6 on her snack is 0.33, find the probability Lucy chooses The Amenic.

Find the probability Lucy chose to go the The Seivom, given that she spent less than $5 on her snack.

Minnie and Dennis are playing a game with a box of black and red counters.  The winner is the first to take two counters of the same colour from the box, without looking.  Dennis goes first and draws both counters at the same time.  There are three more red counters than black counters and the probability of Dennis winning on his first attempt is . 

Find the two possible options for the number of black counters in the box.

Dennis does not win, keeps his two counters and Minnie takes two counters from the box. The probability she draws two red counters, given that she wins is , find the original number of black counters that were in the box.

A bag contains tokens that each have a single number written on them.  The integers between 1 and 15 are all represented on the tokens.  There is one token with a ‘1’ on it, two tokens with a ‘2’ on them, and so on, up until fifteen tokens with a ‘15’ on them. 

A token is taken from the bag and the number on it is recorded, but it is not replaced in the bag.  A second token is then taken from the bag and the number on it is recorded. 

Using a tree diagram, or otherwise, work out the probabilities of the following events:

A crafty coyote spends most of his spare time trying to catch a very fast roadrunner bird.  The coyote’s schemes always involve one of three items procured from a well-known mail order retailer – a crate of TNT (with probability 0.51), a large boulder (with probability 0.29), or a rocket on wheels.

If the coyote uses a crate of TNT there is a 95% chance it will explode at the wrong time and injure the coyote while the roadrunner escapes; otherwise it will simply not explode at all and the roadrunner will escape.  If the coyote uses a large boulder there is an 85% chance it will injure him by landing on his head while the roadrunner escapes; otherwise it will injure the coyote by landing on his foot while the roadrunner escapes.  If the coyote uses a rocket on wheels, there is a 60% chance he will injure himself by running into a cliff face while the roadrunner escapes; otherwise the roadrunner will escape after tricking the coyote into riding the rocket off the top of a cliff.   If the coyote rides the rocket off the top of a cliff he will either get injured when the rocket explodes in mid-air, or get injured by crashing into a mountain on the other side of the valley, or land safely on the ground after his parachute unexpectedly functions properly.  It is twice as likely that the rocket will explode as it is that the coyote will crash into a mountain, and five times as likely that the coyote will crash into a mountain as it is that he will land safely using his parachute.

Draw a tree diagram to represent this information.

Given that the coyote is injured during one of his schemes, find the exact probability that he is not injured by an explosion.

Given that the coyote is not injured during one of his schemes, find the exact probability that his scheme involved a rocket on wheels.