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Probability, MET2 2022 VCAA 3

Mika is flipping a coin. The unbiased coin has a probability of of landing on heads and of landing on tails.

Let be the binomial random variable representing the number of times that the coin lands on heads.

Mika flips the coin five times.

    1. Find .   (1 mark)
    2. Find (1 mark)
    3. Find , correct to three decimal places.   (2 marks)
    4. Find the expected value and the standard deviation for .   (2 marks)

The height reached by each of Mika's coin flips is given by a continuous random variable, , with the probability density function
  


  

where is the vertical height reached by the coin flip, in metres, between the coin and the floor, and and are real constants.

    1. State the value of the definite integral .   (1 mark)
    2. Given that    and  , find the values of and .   (3 marks)

    3. The ceiling of Mika's room is 3 m above the floor. The minimum distance between the coin and the ceiling is a continuous random variable, , with probability density function .
    4. The function is a transformation of the function given by , where is the minimum distance between the coin and the ceiling, and and are real constants.
    5. Find the values of and .   (1 mark)

  1. Mika's sister Bella also has a coin. On each flip, Bella's coin has a probability of of landing on heads and of landing on tails, where is a constant value between 0 and 1 .
  2. Bella flips her coin 25 times in order to estimate .
  3. Let be the random variable representing the proportion of times that Bella's coin lands on heads in her sample.
    1. Is the random variable discrete or continuous? Justify your answer.   (1 mark)
    2. If , find an approximate 95% confidence interval for , correct to three decimal places.   (1 mark)
    3. Bella knows that she can decrease the width of a 95% confidence interval by using a larger sample of coin flips.
    4. If , how many coin flips would be required to halve the width of the confidence interval found in part c.ii.?   (1 mark)

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a. i.        ii.        iii.    (3 d.p.)

a. iv   

b. i.      

b. ii.   

b. iii.

c. i.      ii.      iii.   

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a.i 


 

a.ii  By CAS:        

  
a.iii 

  
By CAS:
 

(3 decimal places)
  

a.iv

  

b.i  


♦♦ Mean mark (b.i) 40%.
MARKER’S COMMENT: Many students did not evaluate the integral or evaluated incorrectly.

b.ii  By CAS:


 


♦♦ Mean mark (b.ii) 47%.
MARKER’S COMMENT: Many students did not give exact answers.

b.iii 

and


♦♦♦♦ Mean mark (b.iii) 10%.
MARKER’S COMMENT: Many students did not attempt this question.

c.i    is discrete.

The number of coin flips must be zero or a positive integer so    is countable and therefore discrete.
 

c.ii 


 

c.iii To halve the width of the confidence interval, the standard deviation needs to be halved.

She would need to flip the coin 100 times


♦♦♦ Mean mark (c.iii) 30%.
MARKER’S COMMENT: Common incorrect answers were 0, 10, 11, 50 and 101.

Statistics, MET2 2020 VCAA 3

A transport company has detailed records of all its deliveries. The number of minutes a delivery is made before or after its schedule delivery time can be modelled as a normally distributed random variable, , with a mean of zero and a standard deviation of four minutes. A graph of the probability distribution of is shown below.
 

  1. If  , find to the nearest minute.   (1 mark)

  2. Find the probability, correct to three decimal places, of a delivery being no later than three minutes after its scheduled delivery time, given that it arrives after its scheduled delivery time.   (2 marks)

  3. Using the model described above, the transport company can make 46.48% of its deliveries over the interval  .
  4. It has an improved delivery model with a mean of and a standard deviation of four minutes.
  5. Find the values of , correct to one decimal place, so that 46.48% of the transport company's deliveries can be made over the interval     (3 marks)

A rival transport company claims that there is a 0.85 probability that each delivery it makes will arrive on time or earlier.

Assume that whether each delivery is on time or earlier is independent of other deliveries.

  1. Assuming that the rival company's claim is true, find the probability that on a day in which the rival company makes eight deliveries, fewer than half of them arrive on time or earlier. Give your answer correct to three decimal places.   (2 marks)

  2. Assuming that the rival company's claim is true, consider a day in which it makes deliveries.
    1. Express, in terms of , the probability that one or more deliveries will not arrive on time or earlier.   (1 mark)

    2. Hence, or otherwise, find the minimum value of such that there is at least a 0.95 probability that one or more deliveries will not arrive on time or earlier.   (1 mark)
  3. An analyst from a government department believes the rival transport company's claim is only true for deliveries made before 4 pm. For deliveries made after 4 pm, the analyst believes the probability of a delivery arriving on time or earlier is , where 
  4. After observing a large number of the rival transport company's deliveries, the analyst believes that the overall probability that a delivery arrives on time or earlier is actually 0.75
  5. Let the probability that a delivery is made after 4 pm be .
  6. Assuming that the analyst's belief are true, find the minimum and maximum values of .   (2 marks)

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  5.  i.
  6. ii.

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a.   


 

b.   
   
   

 

c.   

 

d.   


 

e.i.   


 

e.ii.  

 
 

 

 

f.   

 
 
   

 

Probability, MET2 2021 VCAA 4

A teacher coaches their school's table tennis team.

The teacher has an adjustable ball machine that they use to help the players practise.

The speed, measured in metres per second, of the balls shot by the ball machine is a normally distributed random variable .

The teacher sets the ball machine with a mean speed of 10 metres per second and standard deviation of 0.8 metres per second.

  1. Determine  , correct to three decimal places.   (1 mark)

  2. Find the value of , in metres per second, which 80% of ball speeds are below. Give your answer in metres per second, correct to one decimal place.   (1 mark) 

The teacher adjusts the height setting for the ball machine. The machine now shoots balls high above the table tennis table.

Unfortunately, with the new height setting, 8% of balls do not land on the table.

Let    be the random variable representing the sample proportion of the balls that do not land on the table in random samples of 25 balls.

  1. Find the mean and the standard deviation of  .   (2 marks)

  2. Use the binomial distribution to find  , correct to three decimal places.   (2 marks) 

The teacher can also adjust the spin setting on the ball machine.

The spin, measured in revolutions per second, is a continuous random variable  with the probability density function
 

       
 

  1. Find the maximum possible spin applied by the ball machine, in revolutions per second.   (1 mark)

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  6. This content is no longer in the Study Design.
  7. This content is no longer in the Study Design.
  8. This content is no longer in the Study Design.

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a.   


 

b.   
 

c.   

♦ Mean mark part (c) 45%.


 

d.   

♦ Mean mark part (d) 49%.

 
 
 

 

e.   

♦♦ Mean mark part (e) 21%.

 

f.    This content is no longer in the Study Design.

g.    This content is no longer in the Study Design.

h.    This content is no longer in the Study Design.

Probability, MET2 2021 VCAA 6 MC

The probability of winning a game is 0.25

The probability of winning a game is independent of winning any other game.

If Ben plays 10 games, the probability that he will win exactly four rimes is closest to

  1. 0.1460
  2. 0.2241
  3. 0.9219
  4. 0.0781
  5. 0.7759
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Statistics, MET2 2019 VCAA 4

The Lorenz birdwing is the largest butterfly in Town A.

The probability density function that describes its life span, , in weeks, is given by
 


 

  1. Find the mean life span of the Lorenz birdwing butterfly.  (2 marks)

  2. In a sample of 80 Lorenz birdwing butterflies, how many butterflies are expected to live longer than two weeks, correct to the nearest integer?  (2 marks)

  3. What is the probability that a Lorenz birdwing butterfly lives for at least four weeks, given that it lives for at least two weeks, correct to four decimal places?  (2 marks)

The wingspans of Lorenz birdwing butterflies in Town A are normally distributed with a mean of 14.1 cm and a standard deviation of 2.1 cm.

  1. Find the probability that a randomly selected Lorenz birdwing butterfly in Town A has a wingspan between 16 cm and 18 cm, correct to four decimal places.  (1 mark)

  2. A Lorenz birdwing butterfly is considered to be very small if its wingspan is in the smallest 5% of all the Lorenz birdwing butterflies in Town A.

     

    Find the greatest possible wingspan, in centimetres, for a very small Lorenz birdwing butterfly in Town A, correct to one decimal place.  (1 mark)

Each year, a detailed study is conducted on a random sample of 36 Lorenz birdwing butterflies in Town A.

A Lorenz birdwing butterfly is considered to be very large if its wingspan is greater than 17.5 cm. The probability that the wingspan of any Lorenz birdwing butterfly in Town A is greater than 17.5 cm is 0.0527, correct to four decimal places.

    1. Find the probability that three or more of the butterflies, in a random sample of 36 Lorenz birdwing butterflies from Town A, are very large, correct to four decimal places.  (1 mark)

    2. The probability that or more butterflies, in a random sample of 36 Lorenz birdwing butterflies from Town A, are very large is less than 1%.

       

      Find the smallest value of , where is an integer.  (2 marks)

    3. For random samples of 36 Lorenz birdwing butterflies in Town A, is the random variable that represents the proportion of butterflies that are very large.
    4. Find the expected value and the standard deviation of , correct to four decimal places.  (2 marks)

    5. What is the probability that a sample proportion of butterflies that are very large lies within one standard deviation of 0.0527, correct to four decimal places? Do not use a normal approximation.  (2 marks)

  2. The Lorenz birdwing butterfly also lives in Town B.

     

    In a particular sample of Lorenz birdwing butterflies from Town B, an approximate 95% confidence interval for the proportion of butterflies that are very large was calculated to be (0.0234, 0.0866), correct to four decimal places.

     

    Determine the sample size used in the calculation of this confidence interval.  (2 marks)

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a.   
   
   

 

b.   
   
   

 

 
 

 

c.   
   
   

 

d.   

 

e.   

 

 

f.i.   

 

f.ii.   
 

 

f.iii.   
 

 

f.iv.       

 

g.   

Probability, MET2 2017 VCAA 3

The time Jennifer spends on her homework each day varies, but she does some homework every day.

The continuous random variable , which models the time, in minutes, that Jennifer spends each day on her homework, has a probability density function , where

 

 

  1. Sketch the graph of on the axes provided below.  (3 marks)


     

       
     

  2. Find  .  (2 marks)

  3. Find  .  (2 marks)

  4. Find such that  , correct to four decimal places.  (2 marks)

  5. The probability that Jennifer spends more than 50 minutes on her homework on any given day is . Assume that the amount of time spent on her homework on any day is independent of the time spent on her homework on any other day.

     

    1. Find the probability that Jennifer spends more than 50 minutes on her homework on more than three of seven randomly chosen days, correct to four decimal places.  (2 marks)

    2. Find the probability that Jennifer spends more than 50 minutes on her homework on at least two of seven randomly chosen days, given that she spends more than 50 minutes on her homework on at least one of those days, correct to four decimal places.  (2 marks)

Let be the probability that on any given day Jennifer spends more than minutes on her homework.

Let be the probability that on two or three days out of seven randomly chosen days she spends more than minutes on her homework.

  1. Express as a polynomial in terms of .  (2 marks)

    1. Find the maximum value of , correct to four decimal places, and the value of for which this maximum occurs, correct to four decimal places.  (2 marks)

    2. Find the value of for which the maximum found in part g.i. occurs, correct to the nearest minute.  (2 marks)


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  1.  

Show Worked Solution

a.   

MARKER’S COMMENT: Many did not draw graph along -axis between 0 and 20 and for  .

 

b.   

 

c.   

 

d.   

♦ Mean mark part (d) 36%.

 

 

e.i.   

 

e.ii.   
   
   

 

f.   

♦ Mean mark part (f) 36%.

 
 
   
 

 

g.i.   

♦♦ Mean mark part (g)(i) 30%.

 

 

g.ii.   

♦♦♦ Mean mark part (g)(ii) 8%.

 

 

 

Probability, MET2 2008 VCAA 13 MC

According to a survey, 30% of employed women have never been married.

If 10 employed women are selected at random, the probability (correct to four decimal places) that at least 7 have never been married is

A.   

B.   

C.   

D.   

E.   

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Probability, MET2 2011 VCAA 2*

In a chocolate factory the material for making each chocolate is sent to one of two machines, machine A or machine B.

The time, seconds, taken to produce a chocolate by machine A, is normally distributed with mean 3 and standard deviation 0.8.

The time, seconds, taken to produce a chocolate by machine B, has the following probability density function
 


 

  1. Find correct to four decimal places

    1.    (1 mark)

    2.    (3 marks)

  2. Find the mean of , correct to three decimal places.   (3 marks)

  3. It can be shown that  . A random sample of 10 chocolates produced by machine B is chosen. Find the probability, correct to four decimal places, that exactly 4 of these 10 chocolate took 3 or less seconds to produce.   (2 marks)

All of the chocolates produced by machine A and machine B are stored in a large bin. There is an equal number of chocolates from each machine in the bin.

It is found that if a chocolate, produced by either machine, takes longer than 3 seconds to produce then it can easily be identified by its darker colour.

  1. A chocolate is selected at random from the bin. It is found to have taken longer than 3 seconds to produce.
  2. Find, correct to four decimal places, the probability that it was produced by machine A.   (3 marks)

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a.i. 

a.ii.

b.   

c.   

d.   

Show Worked Solution

a.i.  

 

a.ii.   
   
   

 

b.   
   

 

c.  


 

   
 

♦♦♦ Mean mark part (e) 19%.
MARKER’S COMMENT: Students who used tree diagrams were the most successful.

 

d.   
 
 

Probability, MET2 2016 VCAA 3*

A school has a class set of 22 new laptops kept in a recharging trolley. Provided each laptop is correctly plugged into the trolley after use, its battery recharges.

On a particular day, a class of 22 students uses the laptops. All laptop batteries are fully charged at the start of the lesson. Each student uses and returns exactly one laptop. The probability that a student does not correctly plug their laptop into the trolley at the end of the lesson is 10%. The correctness of any student’s plugging-in is independent of any other student’s correctness.

  1. Determine the probability that at least one of the laptops is not correctly plugged into the trolley at the end of the lesson. Give your answer correct to four decimal places.   (2 marks)

  2. A teacher observes that at least one of the returned laptops is not correctly plugged into the trolley.
  3. Given this, find the probability that fewer than five laptops are not correctly plugged in. Give your answer correct to four decimal places.   (2 marks)

The time for which a laptop will work without recharging (the battery life) is normally distributed, with a mean of three hours and 10 minutes and standard deviation of six minutes. Suppose that the laptops remain out of the recharging trolley for three hours.

  1. For any one laptop, find the probability that it will stop working by the end of these three hours. Give your answer correct to four decimal places.   (2 marks)

A supplier of laptops decides to take a sample of 100 new laptops from a number of different schools. For samples of size 100 from the population of laptops with a mean battery life of three hours and 10 minutes and standard deviation of six minutes,  is the random variable of the distribution of sample proportions of laptops with a battery life of less than three hours.

  1. Find the probability that . Give your answer correct to three decimal places. Do not use a normal approximation.   (3 marks)

It is known that when laptops have been used regularly in a school for six months, their battery life is still normally distributed but the mean battery life drops to three hours. It is also known that only 12% of such laptops work for more than three hours and 10 minutes.

  1. Find the standard deviation for the normal distribution that applies to the battery life of laptops that have been used regularly in a school for six months, correct to four decimal places.   (2 marks)

The laptop supplier collects a sample of 100 laptops that have been used for six months from a number of different schools and tests their battery life. The laptop supplier wishes to estimate the proportion of such laptops with a battery life of less than three hours.

  1. Suppose the supplier tests the battery life of the laptops one at a time.
  2. Find the probability that the first laptop found to have a battery life of less than three hours is the third one.   (1 mark)

The laptop supplier finds that, in a particular sample of 100 laptops, six of them have a battery life of less than three hours.

  1. Determine the 95% confidence interval for the supplier’s estimate of the proportion of interest. Give values correct to two decimal places.   (1 mark)

  2. The supplier also provides laptops to businesses. The probability density function for battery life, (in minutes), of a laptop after six months of use in a business is
     

     


     

  3. Find the mean battery life, in minutes, of a laptop with six months of business use, correct to two decimal places.   (1 mark)

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a.  

 

 
 

 

 b.   

MARKER’S COMMENT: Early rounding was a common mistake, producing 0.9312.

 

c.   

MARKER’S COMMENT: Some working must be shown for full marks in questions worth more than 1 mark.

 

d.   

♦ Mean mark part (d) 33%.

 
 
 

 

e.   

 

f.   
   

 

g.   

 

h.   
 

Probability, MET2 2010 VCAA 12 MC

A soccer player is practising her goal kicking. She has a probability of    of scoring a goal with each attempt. She has 15 attempts.

The probability that the number of goals she scores is less than 7 is closest to

A.  

B.  

C.  

D.  

E.  

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♦ Mean mark 44%.

Probability, MET2 2016 VCAA 17 MC

Inside a container there are one million coloured building blocks. It is known that 20% of the blocks are red.

A sample of 16 blocks is taken from the container. For samples of 16 blocks,  is the random variable of the distribution of sample proportions of red blocks. (Do not use a normal approximation.)

is closest to

A. 

B. 

C. 

D. 

E. 

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Probability, MET2 2015 VCAA 3

Mani is a fruit grower. After his oranges have been picked, they are sorted by a machine, according to size. Oranges classified as medium are sold to fruit shops and the remainder are made into orange juice.

The distribution of the diameter, in centimetres, of medium oranges is modelled by a continuous random variable, , with probability density function
 


 

    1. Find the probability that a randomly selected medium orange has a diameter greater than 7 cm.   (2 marks)

    2. Mani randomly selects three medium oranges.
    3. Find the probability that exactly one of the oranges has a diameter greater than 7 cm.
    4. Express the answer in the form , where and are positive integers.   (2 marks)

  2. Find the mean diameter of medium oranges, in centimetres.   (1 mark)

For oranges classified as large, the quantity of juice obtained from each orange is a normally distributed random variable with a mean of 74 mL and a standard deviation of 9 mL.

  1. What is the probability, correct to three decimal places, that a randomly selected large orange produces less than 85 mL of juice, given that it produces more than 74 mL of juice?  (2 marks)

Mani also grows lemons, which are sold to a food factory. When a truckload of lemons arrives at the food factory, the manager randomly selects and weighs four lemons from the load. If one or more of these lemons is underweight, the load is rejected. Otherwise it is accepted.

It is known that 3% of Mani’s lemons are underweight.

    1. Find the probability that a particular load of lemons will be rejected. Express the answer correct to four decimal places.   (2 marks)

    2. Suppose that instead of selecting only four lemons, lemons are selected at random from a particular load.
    3. Find the smallest integer value of such that the probability of at least one lemon being underweight exceeds 0.5  (2 marks)

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a.i.   
   
   

 

a.ii.  

 

 

 

b.  
   
   

 

c.  

 
 
   
 

 

d.i. 

 

 

 
 
 

 

d.ii.  

♦ Mean mark 44%.

Probability, MET2 2013 VCAA 2

FullyFit is an international company that owns and operates many fitness centres (gyms) in several countries. At every one of FullyFit’s gyms, each member agrees to have his or her fitness assessed every month by undertaking a set of exercises called . There is a five-minute time limit on any attempt to complete and if someone completes in less than three minutes, they are considered fit.

  1. At FullyFit’s Melbourne gym, it has been found that the probability that any member will complete in less than three minutes is This is independent of any member.
  2. In a particular week, 20 members of this gym attemp
    1. Find the probability, correct to four decimal places, that at least 10 of these 20 members will complete in less than three minutes (2 marks)

    2. Given that at least 10 of these 20 members complete in less than three minutes, what is the probability, correct to three decimal places, that more than 15 of them complete in less than three minutes?  (3 marks)

  3. When FullyFit surveyed all its gyms throughout the world, it was found that the time taken by members to complete is a continuous random variable , with a probability density function , as defined below.
     

    1. Find , correct to four decimal places.  (2 marks)

    2. In a random sample of 200 FullyFit members, how many members would be expected to take more than four minutes to complete Give your answer to the nearest integer.  (2 marks)

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a.i.   

a.ii.   

b.i.   

b.ii. 

Show Worked Solution

a.i.  

 

 

a.ii.   
   
   

 

b.i.   
   
   
   

 

b.ii.  

♦ Mean mark 48%.

 
 
 
 

Probability, MET2 2012 VCAA 3

Steve, Katerina and Jess are three students who have agreed to take part in a psychology experiment. Each student is to answer several sets of multiple-choice questions. Each set has the same number of questions, , where is a number greater than 20. For each question there are four possible options A, B, C or D, of which only one is correct.

  1. Steve decides to guess the answer to every question, so that for each question he chooses A, B, C or D at random.
  2. Let the random variable be the number of questions that Steve answers correctly in a particular set.
    1. What is the probability that Steve will answer the first three questions of this set correctly?   (1 mark)

    2. Find, to four decimal places, the probability that Steve will answer at least 10 of the first 20 questions of this set correctly.   (2 marks)

    3. Use the fact that the variance of is to show that the value of is 25.   (1 mark)

  1. The probability that Jess will answer any question correctly, independently of her answer to any other question, is  . Let the random variable be the number of questions that Jess answers correctly in any set of 25.
  2. If  , show that the value of   is .   (2 marks)

  3. From these sets of 25 questions being completed by many students, it has been found that the time, in minutes, that any student takes to answer each set of 25 questions is another random variable, , which is normally distributed with mean and standard deviation .
  4. It turns out that, for Jess, and also .
  5. Calculate the values of and , correct to three decimal places.   (4 marks)

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a.i.   
   

 

a.ii.  

 

 

a.iii.   
 
 
 

 

b.i. & b.ii. 

 

c.  

♦♦♦ Mean mark part (c) 19%.

 

 

d.  

♦♦♦ Mean mark part (d) 19%.

   

MARKER’S COMMENT: Many students didn’t use binomial calculations for  here.