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Binomial Summary
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Learn the conditions for the Binomial Distribution (and assumptions)
Be able to use the Binomial Distribution Formula
Be able to use Binomial Distribution tables
Be able to chance ‘p’ to ‘q’
Calculate the Mean & Standard Deviation for a Binomial Distribution given ‘n’ & π
Combine Binomial Distribution with probability trees, conditional, expected value etc
Be able to estimate the Mean & Standard Deviation from a provided Distribution
Be able to calculate the Mean & Standard Deviation from a provided Distribution
using E(X) and E(X2)
Plants generally produce several flowers each season that need to be pollinated in
order for fruit to be produced. A particular fruit plant produces one flower in one
season per year. The flower must be pollinated in order for a single fruit to be
produced.
This plant type can be pollinated by bees carrying pollen from flower to flower
between plants of the same type. Bee pollination has a 67% successful pollination
rate.
(a) Using an appropriate distribution, calculate the probability that one plant
produces at least three fruit over four years (seasons), if bee pollination is used.
In your answer, you should justify your choice of distribution, identify the
parameters of this distribution, and state any assumptions you make.
(b) This plant type can also be pollinated by hand, with a person transferring pollen
from flower to flower between plants of the same type. Hand pollination has a
91% successful pollination rate.
A home grower has 12 plants of this plant type.
Calculate the probability that for both of two years (seasons), only one of the 12
plants does not produce fruit if hand pollination is used. State any assumptions
you make.
Binomial distribution
n = 4, p = 0.67
Applying this distribution because:
 fixed number of trials (four seasons/years)
 fixed probability (67% success rate for pollination)
 two outcomes (pollinated, not pollinated)
 independence of events (a plant being pollinated or not does
not affect chances of the same plant being pollinated or not
in another season/year).

Assumption is that the bees visit all the plants.
P(X ≥ 3) = 1 – P(X ≤ 2) = 1 – 0.4015 = 0.5985
Binomial distribution
n = 12, p = 0.91
P(X = 11) = 0.3827
P(11 plants produce fruit for both seasons/years)
= 0.38272 = 0.1465
Assuming whether a plant is pollinated or not in one season
does not affect the chances of the same plant being pollinated
or not in another season/year, and assuming whether a plant is
pollinated or not in one season/year does not affect the
chances of another plant being pollinated or not in the same
season/year.
Probability
calculated with
identification of
probability
distribution and
parameters.
Probability
calculated with
identification of
probability
distribution and
parameter
AND
justification of
applying this
distribution
linked to the
context.
Probability
calculated for
one season.
Probability
calculated for
both seasons
with at least one
assumption of
independence
stated.
Statistics and Modelling Probability Distributions 90646, name_______________
2010 Exam ‘A’
The farmer finds that insects have damaged 5% of all the sweet corn cobs grown and therefore these damaged sweet corn cobs will be
unable to be given to the school.
Find the probability that in the next 10 sweet corn cobs the farmer picks, at least 3 cobs will not be able to be given to the school because
of insect damage.
2009 Exam ‘A, M & E’
One day Tom and Tane go surfing at a local beach. Tom manages to ride a wave, on average, 7 out of every 10 times he tries. Tane
manages to ride a wave, on average, 6 out of every 10 times he tries.
Calculate the probability that Tom and Tane each ride a total of 2 of the next 5 waves.
State what probability distribution model you use to solve this problem and justify your choice, including any assumptions you make.
2008 exam ‘A’
Clare sells her chicken eggs to a supermarket. She sends the eggs in boxes each containing 10 cartons. The probability that any carton
contains an egg with a cracked shell is found to be 0.05. If more than two cartons contain cracked shells then the whole box of 10 cartons
is rejected. Assume that eggs with cracked shells occur independently.
Calculate the probability that a randomly selected box is rejected.
2011 Exam ‘A & E’
Lightning strikes affected 16% of all hectares of pine trees planted in New Zealand last year.
A farmer had planted 18 hectares of pine trees.
Assume a lightning strike in one hectare is independent of a lightning strike in another hectare.
Calculate the probability that at least 4 hectares of the farmer’s trees experienced lightning strikes last year.
You must state the distribution used and justify your choice.
2011 Exam ‘A’
A conservationist has permission to enter a pine plantation to capture moreporks and fit leg bands on them so the birds can be studied.
You may assume that moreporks are located randomly and independently throughout the plantation.
70 percent of the capture attempts are successful.
The conservationist locates 4 moreporks on his next trip.
Calculate the probability that at least one of these birds is successfully captured.
Statistics and Modelling Probability Distributions 90646, ANSWERS
2010 Exam
The farmer finds that insects have damaged 5% of all the sweet corn cobs grown and therefore these damaged sweet corn cobs will be
unable to be given to the school.
Find the probability that in the next 10 sweet corn cobs the farmer picks, at least 3 cobs will not be able to be given to the school because
of insect damage.
2009 Exam
One day Tom and Tane go surfing at a local beach. Tom manages to ride a wave, on average, 7 out of every 10 times he tries. Tane
manages to ride a wave, on average, 6 out of every 10 times he tries.
Calculate the probability that Tom and Tane each ride a total of 2 of the next 5 waves.
State what probability distribution model you use to solve this problem and justify your choice, including any assumptions you make.
Binomial :
- Probability is constant at 0.7 for each trial for Tom and 0.6 for Tane.
- There are only two outcomes, catch a wave or not.
- There is a fixed number of trials: 5 waves.
- We must assume that Tane catching a wave is independent of him catching the next wave he tries for, and similarly for
Tom.
2008 exam
Clare sells her chicken eggs to a supermarket. She sends the eggs in boxes each containing 10 cartons. The probability that any carton
contains an egg with a cracked shell is found to be 0.05. If more than two cartons contain cracked shells then the whole box of 10 cartons
is rejected. Assume that eggs with cracked shells occur independently.
Calculate the probability that a randomly selected box is rejected.
2011 Exam (Excellence)
Lightning strikes affected 16% of all hectares of pine trees planted in New Zealand last year.
A farmer had planted 18 hectares of pine trees.
Assume a lightning strike in one hectare is independent of a lightning strike in another hectare.
Calculate the probability that at least 4 hectares of the farmer’s trees experienced lightning strikes last year.
You must state the distribution used and justify your choice.
P ( X ≥ 4) = 1 – 0.67713 = 0.32287
Distribution is binomial. Parameters are
n = 18 and p = 0.16
Fixed number of trials: 18 hectares
Trials success or failure: either lightning strikes or it doesn’t.
Lightning strikes are assumed to be independent of each other.
Probability of a lightning strike is constant at 0.16.
A conservationist has permission to enter a pine plantation to capture moreporks and fit leg bands on them so the birds can be studied.
You may assume that moreporks are located randomly and independently throughout the plantation.
70 percent of the capture attempts are successful.
The conservationist locates 4 moreporks on his next trip.
Calculate the probability that at least one of these birds is successfully captured.
‘Binomial: n = 4, p = 0.7
P(X ≥ 1) = 1 – P(X = 0)
Change to ‘not captured’
P(X = 0) is P(X = 4) with p = 0.3 = 0.0081
P(X ≥ 1) = 1 – 0.0081 = 0.9919 (tables & GC)