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Distributions

Question 1

[Maximum mark: 10]



Consider the following variables. Find which are discrete and which are continuous.

a) The amount of water in an aquarium.


b) The number of clients in a hotel.


c) The final score of a team in a football match.


d) The time it takes to drive to a park.


e) The mass of a book.

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Question 2

[Maximum mark: 6]



Consider the following tables. Are they probability distributions?


a)

x P(x)
1 0.26
2 0.37
3 0.31
4 0.26

b)

x P(x)
100 0.4
110 0.3
120 0.2
130 0.05
140 0.05

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Question 3Register

[Maximum mark: 6]



Consider the following normal distributions.


a) Normal distribution with \( \mu = 5,000 \) and \( \sigma = 250 \). What proportion of the data falls between \( 4,750 \) and \( 5,250 \)? Between \( 4,500 \) and \( 5,500 \)?


b) Normal distribution with \( \mu = 375 \) and variance \( = 225 \). What proportion of the data falls between \( 350 \) and \( 400 \)? Between \( 300 \) and \( 450 \)?

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Question 4Register

[Maximum mark: 8]



A family is planning their summer holiday. They want to go to the beach each time the weather permits. The probability distribution of clear and sunny days is as follows:

x 6 7 8 9 10 11 12 13 14
P(x) 0.04 0.07 0.20 0.22 0.19 0.12 0.08 0.06 0.02

What is the probability that the family will go to the beach:


a) At least 11 times?


b) Fewer than 8 times?


c) Between 8 and 10 times?


d) More than 14 times?

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Question 5Register

[Maximum mark: 7]



There is a regular, circular archery target, with 3 distinct sections. The inner most one is labelled A, then B, and the outermost is C. The probability of hitting each section is given as follows:


Region X Y Z
Probability \(\frac{1}{7}\) \(\frac{3}{7}\) \(\frac{2}{7}\)


a) What is the probability that we hit neither target?


Points are gained when a certain region is hit.


Region X Y Z Miss
Points 10 5 1 \( k \)


b) What is the value of \( k \), given that we have a fair game?

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Question 6Register

[Maximum mark: 12]



A farm produces eggs and has found that their weights follow a normal distribution with mean 48.8g and standard deviation 3.40g


An egg is considered small if its weight is in the lighter 25%, and it is considered big if it is heavier than 75% of them.


a) Sketch a diagram of this weight distribution.


b) What is the maximum weight of a small egg?


c) What is the minimum weight of a big egg?


We take 50 eggs at random from this farm. Determine the expected number of eggs that:


d) Weigh less than 46g.


e) Weigh between 45g and 55g.

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Question 7Register

[Maximum mark: 8]



A YouTuber looks at their viewership statistics. They see that viewers have a 60% chance of watching one of their videos to the end. Each full view brings them a revenue of fifty cents.


If 12 people click on one of their videos:


a) Calculate the expected revenue from that video.


b) Calculate the probability that the YouTuber will make exactly $3.5.


c) Calculate the probability that they will make more than $4.


d) How many people would have to click on the video to expect a revenue of $12?

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Question 8Premium

[Maximum mark: 10]



The savings of people in a city are normally distributed with a mean of $4,000 and a standard deviation of $1,200.


a) If we take two people at random in this city, what is the probability that they both have savings between $2,800 and $5,200?


b) How much money do you need to have to be in the top 10% of wealth in this city? Round your answer to two decimal places.


c) 75% of people have less than \( d \) dollars. Find \( d \). Round your answer to two decimal places.


d) What proportion of people have less than $3,000?

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Question 9Premium

[Maximum mark: 9]



Bob walks to school every day, and he likes to count the number of cars he encounters on the way there. The amount varies on a day-to-day basis according to a Poisson distribution with a mean of 50.


a) What is the probability that Bob encounters more than 60 cars on a day?


b) How many times in a week can Bob expect to encounter over 60 cars?


c) Find the probability that Bob encounters less than 300 cars in a week.

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Question 10Premium

[Maximum mark: 9]



The dimater of watermelons in a particular shop are normally distributed with a mean 12cm with a standard deviation of 1.2cm


A watermelon is chosen at random.


a) Find the probability that:


i. the watermelon has a diameter less than 10cm.

ii. the watermelon has a diameter more than 15cm.


20% of watermelons are known to have a diameter bigger than \( d \).


b) Find the value of \( d \). Round your answer to the nearest integer.


Watermelons with a diameter smaller than 8.5cm are discarded by the shop due to the lack of demand for them.


c) Find the probability that a randomly selected watermelon is discarded due to the lack of demand.

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Question 11Premium

[Maximum mark: 9]



In a small village, 10 people have installed alarm systems from the same company for their homes. This alarm system will work as expected 98% of the time and ring when something is detected.


A series of burglaries happen in the neighborhood. What is the probability that:


a) All 10 alarms will work properly?


b) At least 8 of the alarms will work properly?


A squirrel lives in this village. When he runs through a garden, he will trigger the alarm system of the house 30% of the time. Today, he runs in front of all ten houses with an alarm. What is the probability that:


c) The alarm rings when the squirrel runs in front of one house.


d) The squirrel will make at least 3 alarm systems ring.

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Question 12Premium

[Maximum mark: 9]



Cars in a given car company have various prices with a mean of $25,000 and standard deviation of $3,500.


The car dealership wants to put them into categories based on their prices, where they will be classified as cheap, standard, and expensive. Expensive cars have the price range between $40,000 and $48,000.


a) Find the probability that a randomly selected car will be expensive.


For a randomly selected car of price \( x \), we have that \( P(20000 - q < x < 20000 + q) = 0.6 \).


b) Find the value of \( q \). Round your answer to two decimal places.

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Question 13Premium

[Maximum mark: 8]



A car is driving around a race track, and the time it takes for it to complete a lap is normally distributed, with a mean of 45 seconds and a standard deviation of 3 seconds. The car drives 3 laps around the track, but after each lap, it starts over from the start line with a speed of 0, so all laps are under the same circumstances.


a) Find the mean time for the car to drive the 3 laps.


b) What will be the standard deviation of the 3 laps in total?


c) Another car on the track gets a time of 39 seconds. Find the probability that the next lap will beat their time.

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Question 14Premium

[Maximum mark: 15]



The grades of a math class in high school on a certain test are normally distributed, with a mean of 6, and a standard deviation of 1.


a) Sketch the distribution, and draw in the lines corresponding to 1 and 2 standard deviations above the mean.


b) How many standard deviations is a grade of 5.5 away from the mean?


c) What is the probability that a student gets a grade of 7.5 or better?


d) Answer the following:

i) Find the probability a randomly chosen student scores between a 6 and 8.


ii) Out of a 30 student class, how many people do we expect to score between a 6 and an 8?


e) 75% of students scored less than a grade of \( X \). Find \( X \).

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