State whether X is discrete in the following:
X is a random variable for the length of a crocodile.
X is a random variable for the number of games won by Manchester United in a season.
X is a random variable for the number of television a person owns.
X is a random variable for the weight of a hippopotamus to the nearest kilogram.
X is a random variable for the time taken by a driver to complete one lap of a circuit.
X is a random variable for the length of a queue at a supermarket checkout.
X is a random variable for the number of pears on each pear tree in an orchard.
Out of 100 students sitting an exam, 10 student numbers were drawn randomly for a survey. 35 out of the 100 students are female. Can the number of females in the 10 chosen be represented by a discrete random variable?
A multiple choice test contains 10 questions, each with subparts (a) and (b). The answer to each subpart is awarded a half mark if correct, and zero if incorrect. If a student randomly answers each question, can the number of marks gained on this test be modelled by a discrete random variable?
Explain why the following situations cannot be represented by a discrete random variable:
On a popular TV cooking show, contestants are to randomly choose a blue, red or yellow serviette from a box to split themselves into random cooking teams.
A random number generator generates a real number between 1 and 7 inclusive.
The time taken to download a 55-minute episode of Here Come the Habibs during off-peak times is anywhere between 5 and 13 minutes.
At a local bus stop, the time spent waiting for the next bus during peak hour is up to 10 minutes.
The time between customers using the express checkout at a supermarket is monitored for efficiency purposes. Hence, the operations manager is interested in the checkout operator's waiting time for the next customer to arrive.
Paul arrives at the train station each morning between 8:30 am and 8:45 am and records the time he arrives for one week.
The actual capacity of bottles of orange juice, each advertised as containing 1000 \text{ mL}, are monitored for quality assurance.
An ice-cream shop is monitoring the popularity of its various flavours of ice-cream by analysing the flavour preference of randomly chosen customers.
A photocopier in a busy school breaks on a regular basis.
Can the number of breakdowns occuring in a fortnight be represented by a discrete random variable?
Can the time between breakdowns be represented by a discrete random variable? Explain your answer.
The operations manager at a fast food restaurant is monitoring the number of customers arriving at the drive-through on Sunday. During the 24 hours of monitoring, 360 customers used the drive-through.
Can the number of cars in a randomly chosen 30-minute time period be modelled by a discrete random variable?
Can the time elapsed between customers arriving at the drive-through on the Sunday be modelled by a discrete random variable?
The weights of babies born in a local hospital in the last month have been recorded.
A midwife is interested in the probability that the next baby born would weigh more than 2.4 \text{ kg}. If X represents the weight of the next baby born, is X a discrete random variable? Explain your answer.
The midwife is also interested in the probability that of the next 5 babies born, what number of babies would weigh more than 2.4\text{ kg}. If Y represents the number of babies in the next 5 babies born that weigh more than 2.4 \text{ kg}, is Y a discrete random variable? Explain your answer.
The quality control manager of the installation of a fibre-optic network is monitoring the faults found in the cable being used.
Can the metres between successive faults in the fibre optic cable being analysed be modelled by a discrete random variable?
Can the number of faults found in a randomly chosen 100 \text{ m} length of the fibre optic cable be modelled by a discrete random variable?
Lucy has 10 red tea cups, 10 blue tea cups and 10 green cups in her cupboard. She drinks a lot of tea and these tea cups end up in the dishwasher when she’s used them.
Can the colour of tea cup last placed in the dishwasher be modelled by a discrete random variable?
Can the number of blue tea cups found in the dishwasher at any moment be modelled by a discrete random variable?
A coin is tossed three times and the total number of tails is recorded.
Can this experiment be represented by a discrete random variable?
If X represents the number of possible tails in the three coin tosses, list all the possible outcomes of the experiment.
A random number generator generates an integer between 1 and 6 inclusive. If D represents the integer generated, list all possible outcomes for D.
A student designed a game of chance in which two fair tetrahedral dice (both with faces numbered 1, 2, 3 and 4) were thrown and then the score, X, was calculated from the sum of the numbers appearing uppermost on the dice. List all the possible outcomes of X.
Six cards are drawn from a deck of 52 cards, and the number of kings in the draw is recorded. If X represents the number of kings in the draw, list all the possible outcomes of X.
The time between customers using the express checkout at Coles is monitored for efficiency purposes. The next 8 customers arriving at the checkout are observed. The operations manager monitors how many times the employee checkout operator waits more than 3 minutes for the next customer to arrive, in this group of 8.
If Y represents the number of times the checkout operator waited more than 3 until the next customer arrives, for the next 8 customers, list all the possible outcomes of Y.
The actual capacity of bottles of orange juice, each advertised as containing 1000 \text{ mL}, are monitored for quality assurance. The operations manager is monitoring the number of bottles that are under the stated amount. A local deli receives a shipment of these bottles and opens a box containing 9 bottles.
If Y represents the number of bottles that are under the advertised capacity in the box of 9 bottles, list all the possible outcomes.
The burner on an old gas stove ignites immediately on operation with a probability of 70\%. Someone uses this burner 10 times in one day.
If Y represents the number of times the burner ignites immediately that day, list all the possible outcomes.
There is a surge of students falling ill from the flu in the boarding house of a school. The chance that a student from the boarding house visiting the school nurse has to be referred to a doctor for the flu is 2 in 5. On Monday morning, 8 students from the boarding house visit the school nurse.
If Y represents the number of students from the boarding house being referred to the doctor for the flu on this Monday morning, list all the possible outcomes of Y.
A student is choosing two units to study at university: a language and a science unit. They have 4 languages and 7 science units to choose from.
If they choose one of each, find the total number of combinations of choices.
If Italian is one of the languages they can choose from, find the probability they choose Italian as their language.
French is one of the available languages. Find the probability they choose French as their language given that they choose Chemistry as their science unit.
A netball coach is choosing players for the Goal Keeper and Goal Defence positions out of the following people:
Goal Keeper position: Beth, Amy, Joy, Tara.
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Goal Defence position: Eve, Cara, Daisy, Kim, Liz.
The selection for each position is made independently.
Find the probability the coach will choose Amy and Daisy.
Find the probability the coach will choose Amy or Daisy.
If the coach chooses Joy, find the probability she will choose Kim.
Find the probability the coach will choose Eve given that Beth won’t play with her.
In the Have Sum Fun competition, a teacher needs to make a team of 4 people and another team of 3 people. For the larger team, the teacher has 5 students to choose from, including Jack and Julia. For the smaller team, the teacher has 4 students to choose from, including Alvin. The selections are made independently.
Find the probability both Jack and Alvin are chosen.
Find the probability Julia is chosen if Alvin was chosen.
If Julia won’t go without Jack, find the probability they’re both on the team.
Valentina is creating an exercise plan from a list of 30 exercises. Valentina has a total of 10 cardio exercises, 12 gymnastics exercises and 8 weight exercises. If Valentina wants 6 exercises and one of the chosen exercises is cardio, find the probability that all 6 are cardio.
Two dice are rolled, one after another. Find the probability, in simplest form, of rolling:
A pair of fives, given the first die is a five.
A pair with a sum of 9 or more given that the first die is a four.
Laura is allowed to pack 4 toys for her weekend trip to Grandma’s house. Laura has 6 dolls, 7 cars and 8 teddy bears.
Find the probability Laura takes at least one of each toy.
Find the probability Laura only takes teddy bears.
Given that she took exactly 2 dolls, find the probability she took exactly one car.
If Laura will only take teddy bears and dolls, find the probability she took exactly 3 teddy bears.
Roald is taking 4 books to read on his holiday. He has 3 biographies, 6 novels and 5 non-fiction books to choose from.
Find the probability Roald will take at least one biography.
Given that Roald takes at most one novel, find the probability that he takes exactly one non-fiction book.
Christa is painting 3 bedrooms in her house. Christa has 6 colours to choose from for the 3 bedrooms.
Find the probability that all the bedrooms are different colours.
Find the probability that at least 2 bedrooms are the same colour.
Given that at least two of the bedrooms are the same colour, find the probability that the third bedroom is a different colour.
A card is randomly drawn from a standard 52 pack of cards. Find the probability that it's a jack, queen, king or ace if:
No additional information is known.
We know it's a 10, jack or queen.
We know it's a 9 or a queen.
We know it's not 2, 3, 4 or 5.
Four identical balls labelled 1, 2, 3 and 4 are in a bag. Two balls are randomly drawn from the bag in succession and without replacement. Find the probability, in simplest form, that:
The first ball is labelled 4 and the second ball is labelled 2.
The sum of the numbers on the two balls is 5.
The second ball drawn is 1 given that the sum of the numbers on the two balls is 5.
Two cards are randomly drawn without replacement from a deck of cards numbered from 1 to 20. Find the probability that the second card is:
An even number given that the first card is a 10.
Less than 5 given that the first one is a 14.
A number divisible by 5 given that the first card is a 15.
In a population, 15\% of people have brown eyes, 25\% of people have blonde hair, and 10\% of people have both brown eyes and blonde hair. A person is chosen randomly from the population. Find the probability that they have:
Blonde hair, given that they have brown eyes.
Brown eyes, given that they have blonde hair.
A tour guide is taking groups on holiday and wants to mix up the nationalities of the groups, but she does this at random. Each group has 6 people. On this tour, there are 10 Australians, 7 Americans and 12 Chinese people.
In one group, find the probability that there are 2 people of each nationality.
Given that there are exactly 2 Chinese people in the group, find the probability that there are at least 3 Americans.
Find the probability that there are no Australians in the group given that there are exactly 2 Americans.
At a dessert buffet, a family chooses 5 different dessert plates. There are 6 bowls of ice cream, 3 bowls of pudding and 5 slices of cakes to choose from.
Given that the family chooses exactly two bowls of ice cream, find the probability that only one bowl of pudding was chosen.
Given that the family will choose more ice cream than cake and no pudding, find the probability that the family chose 4 bowls of ice cream.
The local school board needs 5 parents, 2 teachers and 3 local business owners to form a committee. Applications come in from 10 parents including Mr and Mrs Jones, 4 teachers and 6 business owners, including the local baker.
Find the probability that Mrs Jones is on the committee if Mr Jones is on the committee.
Given that the local baker is on the committee, find the probability that Mr or Mrs Jones (but not both) are also on the committee.
Given that Mrs Jones won’t be on the committee without Mr Jones, find the probability that they’re both on the committee.
A teacher is creating a maths test with 10 questions. She has 12 calculus questions, 11 algebra questions and 7 trigonometry questions to choose from.
Given that exactly 6 of her questions are algebra questions, find the probability that there are more calculus questions than trigonometry questions.
Given that there are no trigonometry questions, find the probability that there are at most two calculus questions chosen.
A flight departs from Melbourne to Sydney. The following probabilities were found with regards to departure times and weather:
The probability that the flight departs on time, given the weather is fine in Melbourne is 0.8.
The probability that the flight departs on time, given the weather is not fine in Melbourne is 0.6.
The probability that the weather is fine on any particular day in July is 0.3.
Find the probability that:
The flight from Melbourne to Sydney departs on time in a day in July.
The weather is fine in Melbourne given that the flight departs on time on a day in July.