1. AS 3.3 Probability Starters
Venn Diagrams A Soln
Venn Diagrams B Soln
Probability Trees A Soln
Probability Trees B Soln
Probability Trees C Soln
Probability Table Soln
Expected Value A Soln
Expected Value B Soln
Expected Value C Soln
Expected Value D Soln
Variance A Soln
Variance B Soln
Variance C Soln
Variance D Soln
Arrangements Soln
Permutations Soln
Combinations Soln
Probabilities A Soln
Probabilities B Soln
Conditional Probability Soln
Mixed Problems A Soln
Mixed Problems B Soln
Mixed Problems C Soln

2. Venn Diagrams A
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?

3. Venn Diagrams A Soln
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?

4. Venn Diagrams B
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?

5. Venn Diagrams B Soln
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?

6. Probability Trees A
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?

7. Probability Trees A Soln
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?

8. Probability Trees B
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?

9. Probability Trees B Soln
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?

10. Probability Trees C
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?

11. Probability Trees C Soln
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?

12. Probability Table
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.

13. Probability Table Soln
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.

14. Expected Value A
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)
1) What number is under the spilt curry slug smoothie
2) What is the expected number of potions?

15. Expected Value A Soln
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)
1) What number is under the spilt curry slug smoothie
2) What is the expected number of potions?

16. Expected Value B
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.

17. Expected Value B Soln
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.

18. Expected Value C
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?

19. Expected Value C Soln
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?

20. Expected Value D
Bad Jelly is researching curse effectiveness, recording the different number of curses made in a caldron by her friends.
1) Find the expected number of curses made.   
2) Bad Jelly’s very evil sister Jade 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 Jade’s curses, then doubles the number of curses made by her friends. Find the new expected number of curses.

21. Expected Value D Soln
Bad Jelly is researching curse effectiveness, recording the different number of curses made in a caldron by her friends.
1) Find the expected number of curses made.
2) Bad Jelly’s very evil sister Jade 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 Jade’s curses, then doubles the number of curses made by her friends. Find the new expected number of curses.

22. Variance A
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:
1) What is the average number of newts caught?
2) Calculate the standard deviation of ‘n’

23. Variance A Soln
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:
1) What is the average number of newts caught?
2) Calculate the standard deviation of ‘n’

24. Variance B
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 Jade 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 Jade 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?)

25. Variance B Soln
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 Jade 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 Jade 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?)

26. Variance C
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?

27. Variance C Soln
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?

28. Variance D
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’.

29. Variance D Soln
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’.

30. Arrangements
Bart entertains himself with 4 activities Lets call them ‘A’ ‘B’ ‘C’ and ‘D’
1) On the weekend Bart completes all four activities without repeating. (order makes a difference) How many different arrangements are possible?
 2) After school Bart completes up to four activities without repeating.
How many different arrangements are possible?  
3) Before school he completes up to four activities with repeating allowed.
4) How many ways can the ‘before school’ and ‘after school’ activities be combined together?

31. Arrangements Soln
Bart entertains himself with 4 activities Lets call them ‘A’ ‘B’ ‘C’ and ‘D’
1) On the weekend Bart completes all four activities without repeating. (order makes a difference) How many different arrangements are possible?
 2) After school Bart completes up to four activities without repeating.
How many different arrangements are possible?  
3) Before school he completes up to four activities with repeating allowed.
4) How many ways can the ‘before school’ and ‘after school’ activities be combined together?

32. Permutations
Ralph enjoys eating different glues. He has 14 different glue brands in his collection
1) If he displayed his 5 favourite glues on a shelf.
How many different displays are possible?
2) His ‘flour’ and ‘tree sap’ glues are to feature at the ends of the shelf of 5 glues during conservation week.
3) Ralphs ‘party glue display’ consists of up to 6 different glues in a display cabinet (to attract new friends).
4) If he includes his ‘Ados’ and ‘Super’ glues in his 5 favourite glues on a shelf.

33. Permutations Soln
Ralph enjoys eating different glues. He has 14 different glue brands in his collection
1) If he displayed his 5 favourite glues on a shelf.
How many different displays are possible?
2) His ‘flour’ and ‘tree sap’ glues are to feature at the ends of the shelf of 5 glues during conservation week.
3) Ralphs ‘party glue display’ consists of up to 6 different glues in a display cabinet (to attract new friends).
4) If he includes his ‘Ados’ and ‘Super’ glues in his 5 favourite glues on a shelf.

34. Combinations
Lisa is in a dark mood, but she still laughs at Homers jokes. (Homers memory capacity is 12 jokes)
1) Homer selects 4 jokes to tell
How many different joke selections are possible?
2) Lisa doesn’t want to hear the ‘stick’ joke or the ‘rock’ joke
How many different 4 joke selections are possible?
3) Lisa wants to hear the ‘radiation’ joke and 5 other jokes.
How many different joke selections are possible? 
4) Homer can tell up to 5 jokes before he forgets if he has told a joke already How many different joke selections are possible?

35. Combinations Soln
Lisa is in a dark mood, but she still laughs at Homers jokes. (Homers memory capacity is 12 jokes)
1) Homer selects 4 jokes to tell
How many different joke selections are possible?
2) Lisa doesn’t want to hear the ‘stick’ joke or the ‘rock’ joke
How many different 4 joke selections are possible?
3) Lisa wants to hear the ‘radiation’ joke and 5 other jokes.
How many different joke selections are possible? 
4) Homer can tell up to 5 jokes before he forgets if he has told a joke already How many different joke selections are possible?

36. Probability A
Krusty is far better at remembering jokes than Homer. (Krusty can remember 18 jokes & the order of jokes is unimportant)
1) What is the probability that Krusty tells his ice-cream joke and tree joke in a Krusty Show consisting of 8 jokes?
2) What is the probability that Krusty does not tell his car joke, pie joke, & sheep joke in a 10 joke Krusty Show?
3) What is the probability that Krusty tells his ice-cream joke and not the tree joke in the same Krusty Show consisting of 8 jokes?
4) What is the probability that the car joke is included in a Krusty Show of up to 5 jokes? 

37. Probability A Soln
Krusty is far better at remembering jokes than Homer. (Krusty can remember 18 jokes & the order of jokes is unimportant)
 1) What is the probability that Krusty tells his ice-cream joke and tree joke in a Krusty Show consisting of 8 jokes?
2) What is the probability that Krusty does not tell his car joke, pie joke, & sheep joke in a 10 joke Krusty Show?
3) What is the probability that Krusty tells his ice-cream joke and not the tree joke in the same Krusty Show consisting of 8 jokes?
4) What is the probability that the car joke is included in a Krusty Show of up to 5 jokes? 

38. Probability B
Itchy & Scratchy make up wrestling routines out of a possible 14 ‘holds’ for a TV show (‘holds’ are not repeated)
1) What is the probability that a routine of 6 holds has the ‘nose hold’ first, and the ‘ear hold’ last?
2) What is the probability that a 5 hold show contains the three leg holds?
3) What is the probability that the toe hold and the eyebrow hold occur together in a 10 hold show?
4) What is the probability that a 8 hold show does not have the three leg holds together? 

39. Probability B Soln
Itchy & Scratchy make up wrestling routines out of a possible 14 ‘holds’ for a TV show (‘holds’ are not repeated)
 1) What is the probability that a routine of 6 holds has the ‘nose hold’ first, and the ‘ear hold’ last?
2) What is the probability that a 5 hold show contains the three leg holds?
3) What is the probability that the toe hold and the eyebrow hold occur together in a 10 hold show?
4) What is the probability that a 8 hold show does not have the three leg holds together? 

40. Conditional Probability
Homer has different mental states.
depending on how many meals he has eaten.
What is the probability he has 3 meals given that he has at least one meal?
If Homer has less than 4 meals what is the probability he has 1 meal?
Number of meals 1 2 3
4 or more
P(M = m)
0.45
0.22
0.16
0.17

41. Conditional Probability Soln
Homer has different mental states.
depending on how many meals he has eaten.
What is the probability he has 3 meals given that he has at least one meal?
If Homer has less than 4 meals what is the probability he has 1 meal?
Number of meals 1 2 3
4 or more
P(M = m)
0.45
0.22
0.16
0.17

42. Mixed Problems A
The probability that cat in the hat eats a mouse is 0.46
The probability that the mouse nibbles the cats hat is 0.38
The chance of both happening is 0.22
Find the probability of:
1) The cat not eating a mouse and getting his hat nibbled.
2) Are the events independent? And why?
3) Neither event happening (no eating or nibbling).
4) The cat eating the mouse given that his hat was
nibbled.

43. Mixed Problems A Soln
The probability that cat in the hat eats a mouse is 0.46
The probability that the mouse nibbles the cats hat is 0.38
The chance of both happening is 0.22
Find the probability of:
1) The cat not eating a mouse and getting his hat nibbled.
2) Are the events independent? And why?
3) Neither event happening (no eating or nibbling).
4) The cat eating the mouse given that his hat was
nibbled.