Normal Distribution Starters Binomial A Solns Binomial B

Normal Distribution Starters Binomial A Solns Binomial B Solns Binomial C Solns Poisson A Solns Poisson B Solns Poisson C Solns Poisson D Solns Normal Dist A Solns (z values) Normal Dist B Starter B Solns A Starter B Solns B Normal Dist C Starter C: Solns 1 Starter C: Solns 2 Normal Dist D Starter D: Solns 1 Starter D: Solns 2 Normal Dist E Starter E: Solns 1 Starter E: Solns 2 Normal Dist F Starter F: Solns 1 Starter F: Solns 2 Normal Dist G Starter G: Solns 1 Starter G: Solns 2 Normal Dist H Starter H Solns (inverse z values) Normal Dist I Starter I: Solns 1 Starter I: Solns 2 Mixed problems A Soln Mixed problems B Soln Mixed problems C Soln

1 st Page Binomial A 1) On average one bowl in every 4 has lumpy porridge. If big daddy B has 6 bowls of porridge, find the probability: (a) There are exactly 5 bowls with lumpy porridge. (b) There at most 1 bowl with lumpy porridge. 2) Goldilocks has nightmares on 4 nights each week on average. a) Find the probability of her having more than 2 nights with nightmares in a week. b) Given that she had less than 5 nightmare nights in a week, find the probability that she had only 1 nightmare night.

1 st Page Binomial A Soln 1) On average one bowl in every 4 has lumpy porridge. If big daddy B has 6 bowls of porridge, find the probability: (a) There are exactly 5 bowls with lumpy porridge. (b) There at most 1 bowl with lumpy porridge. 2) Goldilocks has nightmares on 4 nights each week on average. a) Find the probability of her having more than 2 nights with nightmares in a week. b) Given that she had less than 5 nightmare nights in a week, find the probability that she had only 1 nightmare night.

1 st Page Binomial B 1) It is known that 60% of candidates will achieve in an examination. If 5 people sit the examination find the probability: (a) Exactly 3 candidates will achieve. (b) At least 3 candidates will achieve. 2) A drug is known to be 90% effective when it is used to cure a disease. If 20 people are given the drug then X is binomial with n = 20, = 0. 9. a) Find the mean on the binomial distribution b) Find the standard deviation on the binomial distribution

1 st Page Binomial B Soln 1) It is known that 60% of candidates will achieve in an examination. If 5 people sit the examination find the probability: (a) Exactly 3 candidates will achieve. (b) At least 3 candidates will achieve. 2) A drug is known to be 90% effective when it is used to cure a disease. If 20 people are given the drug then X is binomial with n = 20, = 0. 9. a) Find the mean on the binomial distribution b) Find the standard deviation on the binomial distribution

1 st Page Binomial C 1) List the conditions of the binomial distribution. 2) It is known that 18. 5 % of people can turn their eyelids inside out. In a group of 20 people what is the probability that more than 1 person can turn their eyelids inside out?

1 st Page Binomial C Soln 1) Binomial distribution occurs when: (a) There is a fixed number (n) of trials. (b) The result of any trial can be classified as a “success” or a “failure” (c) The probability of a success ( or p) is constant from trial to trial. (d) Trials are independent. P(X =x) = n. Cx x(1 - )n-x 2) It is known that 18. 5 % of people can turn their eyelids inside out. In a group of 20 people what is the probability that more than 1 person can turn their eyelids inside out?

1 st Page Poisson A 1) List the conditions of the Poisson distribution. 2) In a particular marine reserve there are on average 1. 25 crayfish per m 2 of seafloor. In a 20 m 2 area what is the probability that there are 10 crayfish?

1 st Page Poisson A Soln 1) Poisson distribution occurs when: (a) Trials are independent. (b) The events cannot occur simultaneously (c) Events are random and unpredictable (d) The probability of an event occurring is proportional to the interval length (for small intervals) 2) In a particular marine reserve there are on average 1. 25 crayfish per m 2 of seafloor. In a 20 m 2 area what is the probability that there are 10 crayfish?

1 st Page Poisson B 1) Goldilocks breaks three chairs per hour at school because she is over weight. What is the probability that she breaks no more than four chairs in an hour. (Assume chair breakages are independent ) 2) Goldilocks has on average five tantrums per hour. What is the probability that she has at least two tantrums in a given fifteen minute interval. (Assume tantrums are independent )

1 st Page Poisson B Soln 1) Goldilocks breaks three chairs per hour at school because she is over weight. What is the probability that she breaks no more than four chairs and an hour. (Assume chair breakages are independent ) 2) Goldilocks has on average five tantrums per hour. What is the probability that she has at least two tantrums in a given fifteen minute interval. (Assume tantrums are independent )

1 st Page Poisson C 1) Baby bear cries four times a week on average. What is the mean and variance of the number of times he cries and a given day. 2) The probability that baby bears chair is not broken in a given week is 0. 44 a) What is the average number of times the chair is broken in a week. b) What is the standard deviation of the number of times the chair is broken in a week. c) What is the probability that baby bears chair is broken twice in a week.

1 st Page Poisson C Solns 1) Baby bear cries four times a week on average. What is the mean and variance of the number of times he cries and a given day. 2) The probability that baby bears chair is not broken in a given week is 0. 44 a) What is the average number of times the chair is broken in a week. b) What is the standard deviation of the number of times the chair is broken in a week. c) What is the probability that baby bears chair is broken twice in a week.

1 st Page Poisson D The bear hunting season is five months long. On average the bear family get angry four times a month in the bear hunting season and three times a month in the off-season. 1) What is the probability that the bear family get angry twice in the next month? 2) If the bear family did not get angry last month, what is the probability that it is the bear hunting season?

1 st Page Poisson D Soln The bear hunting season is five months long. On average the bear family get angry four times a month in the bear hunting season and three times a month in the off-season. 1) What is the probability that the bear family get angry twice in the next month? 2) If the bear family did not get angry last month, what is the probability that it is the bear hunting season?

1 st Page ‘Z’ values Look up these ‘z’ values to find the corresponding probabilities 1) P(0 < z < 1. 4) = 2) P(0 < z < 2. 04) = 3) P(0 < z < 1. 55) = 4) P(0 < z < 2. 125) = 5) P(-0. 844 < z < 0) = 6) P(-2. 44 < z < 2. 44) = 7) P(-0. 85 < z < 1. 646) = 8) P( z < 2. 048) = 9) P(1. 955 < z < 2. 044) = 10) P( z < -2. 111) = 0 z

1 st Page ‘Z’ value Solutions Look up these ‘z’ values to find the corresponding probabilities 1) P(0 < z < 1. 4) = 2) P(0 < z < 2. 04) = 3) P(0 < z < 1. 55) = 4) P(0 < z < 2. 125) = 5) P(-0. 844 < z < 0) = 6) P(-2. 44 < z < 2. 44) = 7) P(-0. 85 < z < 1. 646) = 8) P( z < 2. 048) = 9) P(1. 955 < z < 2. 044) = 10) P( z < -2. 111) = 0 z

1 st Page Starter B A salmon farm water tank contains fish with a Mean length of 240 mm Calculate the probability of the following (Std dev = 15 mm) 1) P(A fish is between 240 and 250 mm long) = 240 mm 2) P(A fish is between 210 and 260 mm long) = 240 mm 3) P(A fish is less than 254 mm long) = 240 mm 4) P(A fish is less than 220 mm long) = 240 mm 5) P(A fish is between 255 and 265 mm long) = 240 mm

1 st Page Starter B Solns 1 A salmon farm water tank contains fish with a Mean length of 240 mm Calculate the probability of the following (Std dev = 15 mm) 1) P(A fish is between 240 and 250 mm long) = 240 mm 2) P(A fish is between 210 and 260 mm long) = 240 mm 3) P(A fish is less than 254 mm long) = 240 mm

1 st Page Starter B Solns 2 A salmon farm water tank contains fish with a Mean length of 240 mm Calculate the probability of the following (Std dev = 15 mm) 4) P(A fish is less than 220 mm long) = 240 mm 5) P(A fish is between 255 and 265 mm long) = 240 mm

1 st Page Starter C A west coast population of mosquitoes have a Mean weight of 4. 8 kg Calculate the probability of the following (Std dev = 0. 6 kg) 1) What is the probability a mosquito is between 4. 8 kg and 5. 8 kg? 2) What percentage of mosquitoes are between 4 kg and 5 kg? 3) Out of a sample of 120 mosquitoes, how many would be over 6 kg? 4) What percentage of mosquitoes are between 3 kg and 4 kg? 5) What percentage of mosquitoes are under 5. 5 kg? 4. 8 kg

Starter C: Solns 1 1 st Page A west coast population of mosquitoes have a Mean weight of 4. 8 kg Calculate the probability of the following (Std dev = 0. 6 kg) 1) What is the probability a mosquito is between 4. 8 kg and 5. 8 kg? 4. 8 kg 2) What percentage of mosquitoes are between 4 kg and 5 kg? 4. 8 kg

Starter C: Solns 2 1 st Page A west coast population of mosquitoes have a Mean weight of 4. 8 kg Calculate the probability of the following (Std dev = 0. 6 kg) 3) Out of a sample of 120 mosquitoes, how many would be over 6 kg? 4. 8 kg 4) What percentage of mosquitoes are between 3 kg and 4 kg? 4. 8 kg 5) What percentage of mosquitoes are under 5. 5 kg? 4. 8 kg

1 st Page Starter D A room contains flies with a Mean weight of 3. 6 g and a Standard Deviation of 0. 64 kg 1) What is the probability a fly is between 3. 6 g and 5. 8 g? 2) What percentage of flies are between 3 g and 5 g? 3) Out of a sample of 40 flies, how many would be under 4 g? 4) What percentage of flies are between 2 g and 3 g? 5) What percentage of flies are under 2. 5 g 3. 6 g

Starter D: Solns 1 1 st Page A room contains flies with a Mean weight of 3. 6 g and a Standard Deviation of 0. 64 kg 1) What is the probability a fly is between 3. 6 g and 5. 8 g? 3. 6 g 2) What percentage of flies are between 3 g and 5 g? 3. 6 g

Starter D: Solns 2 1 st Page A room contains flies with a Mean weight of 3. 6 g and a Standard Deviation of 0. 64 kg 3) Out of a sample of 40 flies, how many would be under 4 g? 3. 6 g 4) What percentage of flies are between 2 g and 3 g? 3. 6 g 5) What percentage of flies are under 2. 5 g 3. 6 g

1 st Page Starter E The mean weight of a loader scoop of coal is 1. 25 tonnes and a standard deviation of 280 kg 1) What percentage of scoops are between 1. 3 and 1. 5 tonnes? 2) What percentage of scoops are less than 1 tonne? 3) Out of a sample of 500 scoops, how many would be over 1. 4 tonnes? 4) What percentage of scoops are more than 1. 6 tonnes? 1. 25 t 5) What percentage of scoops are between 1 tonne and 2 tonnes

Starter E: Solns 1 1 st Page The mean weight of a loader scoop of coal is 1. 25 tonnes and a standard deviation of 280 kg 1) What percentage of scoops are between 1. 3 and 1. 5 tonnes? 1. 25 t 2) What percentage of scoops are less than 1 tonne? 1. 25 t

Starter E: Solns 2 1 st Page The mean weight of a loader scoop of coal is 1. 25 tonnes and a standard deviation of 280 kg 3) Out of a sample of 500 scoops, how many would be over 1. 4 tonnes? 1. 25 t 4) What percentage of scoops are more than 1. 6 tonnes? 1. 25 t 5) What percentage of scoops are between 1 tonne and 2 tonnes 1. 25 t

1 st Page Starter F The weight of glue paste Ralph eats in a day is normally distributed with a mean of 4. 6 kg & standard deviation of 1. 3 kg 1) How many days in November will Ralph eat less than 5. 5 kg of glue paste? 2) What percentage of days does he eat less than 4 kg of glue paste? 3) Ralph vomits when he eats more than 6 kg of glue in a day. What is the chance of this happening? 4. 6 kg 4) What percentage of days does he eat between 4. 2 kg and 5 kg of glue? 5) What is the probability he does not eat between 3. 5 & 5 kg of glue paste?

Starter F: Solns 1 1 st Page The weight of glue paste Ralph eats in a day is normally distributed with a mean of 4. 6 kg & standard deviation of 1. 3 kg 1) How many days in November will Ralph eat less than 5. 5 kg of glue paste? 4. 6 kg 2) What percentage of days does he eat less than 4 kg of glue paste? 4. 6 kg

Starter F: Solns 2 1 st Page The weight of glue paste Ralph eats in a day is normally distributed with a mean of 4. 6 kg & standard deviation of 1. 3 kg 3) Ralph vomits when he eats more than 6 kg of glue in a day. What is the chance of this happening? 4. 6 kg 4) What percentage of days does he eat between 4. 2 kg and 5 kg of glue? 4. 6 kg 5) What is the probability he does not eat between 3. 5 & 5 kg of glue paste? 4. 6 kg

1 st Page Starter G Itchy & Scratchy have a hammer collection which is normally distributed with a mean of 10. 4 kg & standard deviation of 2. 3 kg 1) What percentage of the hammers weigh less than 8 kg? 2) What is the probability a hammer weighs between 11 kg & 14 kg? 10. 4 kg 3) Scratchy’s head splits open if the hammer is more than 15 kg. What is the chance of this happening? 4) A truck is loaded with 200 hammers. How many of these would be 12 kg or less? 5) 90% of hammers weigh more than what weight?

1 st Page Starter G: Solns 1 Itchy & Scratchy have a hammer collection which is normally distributed with a mean of 10. 4 kg & standard deviation of 2. 3 kg) 1) What percentage of the hammers weigh less than 8 kg? 10. 4 kg 2) What is the probability a hammer weighs between 11 kg & 14 kg? 10. 4 kg

1 st Page Starter G: Solns 2 Itchy & Scratchy have a hammer collection which is normally distributed with a mean of 10. 4 kg & standard deviation of 2. 3 kg) 3) Scratchy’s head splits open if the hammer is more than 15 kg. What is the chance of this happening? 10. 4 kg 4) A truck is loaded with 200 hammers. How many of these would be 12 kg or less? 10. 4 kg 5) 90% of hammers weigh more than what weight? 10. 4 kg

Inverse ‘Z’ values 1 st Page Look up these probabilities to find the corresponding ‘z’ values 1) 2) 0. 3 3) 0. 85 5) 6) 7) 8) 4) 0. 45 0. 08 0. 12 0. 65 0. 02 0 z 0. 4

1 st Page Inverse ‘Z’ values: Solns Look up these probabilities to find the corresponding ‘z’ values 1) 2) 0. 3 3) 0. 85 5) 6) 7) 8) 4) 0. 45 0. 08 0. 12 0. 65 0. 02 0 z 0. 4

1 st Page Starter I Kenny is practicing to be in a William Tell play. He suffers some blood loss which is normally distributed with a mean of 120 m. L & standard deviation of 14 m. L 1) What is the probability his blood loss is less than 100 m. L? 120 m. L 2) 80% of the time his blood loss is more then ‘M’ m. L. Find the value of ‘M’ 3) Kenny passes out when his blood loss is too much. This happens 5% of the time. What is the maximum amount of blood loss Kenny can sustain? 4) 30% of the time Kenny is not concerned by his blood loss? What is his blood loss when he starts to be concerned? 5) The middle 80% of blood losses are between what two amounts?

1 st Page Starter I: Solns 1 Kenny is practicing to be in a William Tell play. He suffers some blood loss which is normally distributed with a mean of 120 m. L & standard deviation of 14 m. L 1) What is the probability his blood loss is less than 100 m. L? 120 m. L 2) 80% of the time his blood loss is more then ‘M’ m. L. Find the value of ‘M’ 120 m. L 3) Kenny passes out when his blood loss is too much. This happens 5% of the time. What is the maximum amount of blood loss Kenny can sustain? 120 m. L

1 st Page Starter I: Solns 2 Kenny is practicing to be in a William Tell play. He suffers some blood loss which is normally distributed with a mean of 120 m. L & standard deviation of 14 m. L 4) 30% of the time Kenny is not concerned by his blood loss? What is his blood loss when he starts to be concerned? 120 m. L 5) The middle 80% of blood losses are between what two amounts? 120 m. L Lucky Kenny is not involved in the knife catching competition!

1 st Page Mixed Problems A 1) Mean maximum temperature is 26°C Standard deviation = 5°C Temp measured to nearest degree. a) P(Temperature at least 30°C) b) P(Temperature between 20 and 30°C) c) P(Temperature between 20 and 24°C inclusive) 2) On average there are 4 frogs per litre of swamp water. What is the probability there are less than 4 frogs in a 2 litre bucket of swamp water

1 st Page Mixed Problems A Soln 1) Mean maximum temperature is 26°C Standard deviation = 5°C Temp measured to nearest degree. a) P(Temperature at least 30°C) b) P(Temperature between 20 and 30°C) c) P(Temperature between 20 and 24°C inclusive) 2) On average there are 4 frogs per litre of swamp water. What is the probability there are less than 4 frogs in a 2 litre bucket of swamp water

1 st Page Mixed Problems B The mean weight of a cake is 1. 5 kg with standard deviation of 0. 34 kg Homer weighs 120 kg. 1) If Homer ate six cakes what is the mean and standard deviation of his total weight. 2) What is the probability Homer weighs over 130 kg?

1 st Page Mixed Problems B Soln The mean weight of a cake is 1. 5 kg with standard deviation of 0. 34 kg Homer weighs 120 kg. 1) If Homer ate six cakes what is the mean and standard deviation of his total weight. 2) What is the probability Homer weighs over 130 kg?

1 st Page Mixed Problems C Each day Homer is exposed to radiation for six minutes on average (variance of 1. 5 minutes) Radiation exposure is considered dangerous if it is for more than five minutes per day. 1) What is the probability that Homer is exposed to dangerous levels of radiation on two consecutive days 2) What is the probability that Homer is exposed to dangerous levels of radiation for at least three days in a five day week

1 st Page Mixed Problems C Soln Each day Homer is exposed to radiation for six minutes on average (variance of 1. 5 minutes) Radiation exposure is considered dangerous if it is for more than five minutes per day. 1) What is the probability that Homer is exposed to dangerous levels of radiation on two consecutive days 2) What is the probability that Homer is exposed to dangerous levels of radiation for at least three days in a five day week