Download 1_Starters - Nayland Maths

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Venn Diagrams A
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A survey of customers at a wild foods fish-n-chip stall resulted in 64%
of customers purchasing boiled worms, 75% purchasing fried slugs,
and 55% purchasing both boiled worms and fried slugs
1) Are purchasing worms and purchasing
fried slugs independent? Why?
2) What is the probability of purchasing
worms, but no fried slugs?
3) What is the probability of purchasing
neither worms nor fried slugs?
4) Does purchasing worms increase or decrease
the probability of purchasing fried slugs? Why?
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Venn Diagrams B
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Bad Jelly the witch falls off her broomstick 15% of all flights. She is
late on 24% of her flights. Strangely these events are independent –
after all, she is a witch.
1) What is the probability that she falls
off her broomstick and is also late?
2) What is the probability that she is late
and does not fall off her broomstick?
3) What percentage of flights involve at least
one of these embarrassing incidents?
4) Attending a caldron conference requires a
witch to be on time and not fall of their
broomstick.
What percentage of conferences
can Bad Jelly attend?
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Probability Trees A
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Bad Jelly attends broomstick flight lessons. Her broomstick is getting
old and fails to take off on 32% of flight lessons. She also crashes on
16% of landing attempts.
1) What is the probability that a flight lesson ends with a crash landing?
2) What is the probability of ‘failing’ a flight lesson?
3) 75% of crashes result in awkward nasal injuries for Bad Jelly
because of her prominent proboscis. What percentage of broomstick
lessons result in nasal injuries?
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Probability Trees B
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After successful broomstick lessons Bad Jelly enters a broomstick drag
racing competition. Bad Jelly is bad and takes an anabolic potion
before the event (82% of anabolic potions work). All entrants undergo
potion testing, but the potion test is incorrect in 16% of results.
(anabolic potion is undetectable if it doesn’t work)
1) What is the chance that Bad Jelly tests positive in the competition?
2) What is the probability that Bad Jelly has the anabolic potion work
and she tests negative in the potion test?
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Probability Trees C
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Bad Jelly is lonely so she decides to go on a date with a
warlock. Due to her prominent proboscis Bad Jelly is
rejected by 85% of warlocks she asks out.
If rejected she immediately shaves and applies a special ‘randy newt’
aftershave potion – guaranteed to get a date 45% of the time – and
then asks again.
Bad Jelly is also very persistent, and by using a ‘time warp’ potion she
is able to ask an infinite number of warlocks for a date.
What is the probability Bad Jelly gets a date without the assistance of a
potion?
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Probability
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Bad Jelly’s potion can have a range of contents and either it is
successful or it explodes.
The probability of:
1) A potion containing a frog.
2) An exploding worm potion.
3) A potion exploding.
4) An exploding potion containing a slug.
5) A worm potion being a success.
Expected Value A
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During a potion making competition Bad Jelly records the number of
potions made by the other witches (Bad Jelly used to be a
mathematician before a terrible differentiation accident)
Number of
potions made
1
2
3
P(X = x)
0.1
0.23
0.34
1) What number is under the spilt curry slug smoothie
2) What is the expected number of potions?
4
5
0.12
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Expected Value B
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Bad Jelly has a set of 8 party potions ready for the Hags & Hooters all
night Rave – the highlight of the witches’ social calendar. Being a bad
(and somewhat desperate) witch she decides to take 3 of the potions
with her. However the evil potion dealer has ‘cut’ the potions and only
4 actually work.
Find the expected number of potions that will work for Bad Jelly at the
Hags & Hooters rave.
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Expected Value C
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Bad Jelly decides to get back at the evil potion dealer by putting curses
on various parts of his anatomy. She brews up 3 different evil curses.
However in her haste she has miss-labelled the curses.
What is the expected number of curses with correct labels?
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Expected Value D
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Bad Jelly is researching curse effectiveness, recording the different
number of curses made in a caldron by her friends.
Number of curses made c
1
2
3
4
5
P(C = c)
0.42
0.32
0.12
0.08
0.06
1) Find the expected number of curses made.
2) Bad Jelly’s very evil sister Kate arrives late (again) and proceeds to
add 3 curses to everyone’s caldron – cos she can. Find the new
expected number of curses made.
3) Bad Jelly does not approve of this rudeness so she removes evil
Kate’s curses, then doubles the number of curses made by her friends.
Find the new expected number of curses.
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Variance A
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Bad Jelly wants to improve the consistency of her ‘randy newt’ potions.
She catches families of newts for the potions. The probabilities of
finding different newt family sizes are given below:
Number of newts ‘n’
1
2
3
4
5
P(N = n)
0.2
0.34
0.24
0.16
0.06
1) What is the average number of newts caught?
2) Calculate the standard deviation of ‘n’
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Variance B
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The number of warts on Bad Jelly’s nose change daily, (each day is
independent) with a mean μ of 5.2 and standard deviation σ of 1.3
warts per day. Bad Jelly’s sister Kate is so evil that only a few warts
can survive on her nose (μ = 3.4 and σ = 0.8)
One day Bad Jelly and evil Kate count their warts.
1) What is the average total number of warts?
2) What is the standard deviation of the total number of warts?
3) Bad Jelly records the number of warts on her own nose for a week.
What was her expected total (and the standard deviation?)
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Variance C
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Bad Jelly is concerned by her wart problem so she visits the STI clinic
(Spells, Tonics & Incantations) and buys two anti-wart potions. The
average number of warts a tonic can treat is 5 (variance = 1.4)
Bad Jelly has many warts, so she applies a spell to make one potion
three times as strong.
1) What is the expected number of warts the potion can now treat?
(And variance?)
Bad Jelly doubles the strength of the other potion, and then combines
the two potions together.
2) What is the expected number of warts the potion can now treat?
(And variance?)
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Variance D
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Bad Jelly dislikes the spotted twerp Harry Potter making money in the
movie business. After the 3rd movie Harry was so unpopular that he
had to ‘buy’ friends by having lots of fast broomsticks to loan out.
On average he has 18 broomsticks (variance = 3.4)
Bad Jelly uses an unstable broomstick destruction curse, which works
on 12 broomsticks on average (variance 4.4)
1) Find the expected number of ‘friends’ Harry Potter will now have.
2) Find the standard deviation of the number of Harry Potter ‘friends’.