Transcript Binomials Ch 4 part 2

Lesson Objective
Be able to calculate probabilities for Binomial situations
Begin to recognise the conditions necessary for a Random variable to have a
Binomial distribution and begin to calculate probabilities using the Binomial
formula.
I toss a biased coin 3 times. The probability that I flip a head is 1/3
Define the Random Variable X = The number of Heads
Draw a probability distribution table for this Random Variable:
Consider the following probability question:
I have a fair spinner (as shown)
The probability that I get a Red is 1/4
P(Red) = 1/4
X = The number of times I spin a Red in 5 spins
Spins:
What will the probability distribution table look like?
5
Consider the following probability question:
The probability that I get a Red is ……
I spin the spinner …. times and count the number of Reds that I get.
What outcomes can I get? What is the probability of each outcome?
P(0 Reds)
=
P(1 Red)
=
P(2 Reds)
=
P(3 Reds)
=
P(4 Reds)
=
P(5 Reds)
=
P(6 Reds)
=
P(7 Reds)
=
P(8 Reds)
=
P(Red) = ……
Spins:
.……
In general if you repeat an experiment ‘n’ times and the probability of success remains
fixed for each. Then you can work out the probability of there being ‘r’ success using the
formula:
P(‘r’ successes) = pr × (1-p)n-r × nCr
Situations that we can use this formula for are called Binomial Situations
And the ‘shape’ of the relating probabilities is called a Binomial Distribution.
Eg. The probability that Ed gets full marks on his homework is 1/5
Assuming that all homework’s have the same level of difficulty, what
is the probability that Ed gets full marks on exactly 3 out of the 8 homeworks?
1) I roll a fair die 8 times. What is the probability that I get 3 sixes?
2) A biased coin has the probability of getting a head as 1/3. I toss the coin
5 times, what is the probability that I get 3 heads?
3) A factory produces ‘widgets’. The probability that a widget is faulty is 10%.
If I check 10 widgets:
a) What is the probability that 3 are faulty?
b) What is the probability that 0 are faulty?
c) What is the probability that less than 4 are faulty?
What are the characteristics of a Random
Variable/situation that has a Binomial Distribution?
Lesson Objective
Understand the characteristics that a situation must have in order to
be modelled using a Binomial distribution
Understand the notation connected with a Binomial distribution
question and be able to use the Binomial distribution to solve a range
of probability problems
What are the characteristics of a Random
Variable/situation that has a Binomial Distribution?
What are the characteristics of a Random
Variable/situation that has a Binomial Distribution?
You have ‘n’ trials.
Each independent and each with a two outcomes, success and failure.
The probability of success ‘p’ remains fixed for each trial.
The Random Variable, X, counts the number of successes in ‘n’ trials.
Then we say that X~B(n,p)
P(X=r) = nCr pr (1-p)n-r
Which of these situations describes a Binomial distribution:
1) In a hospital the number and sex of babies is recorded as they are born. On a
particular day 30 Babies are born. X = The number of boy babies born on that day.
2) I roll a biased die 12 times. X = The number of times I get a score below 5.
3) A bag contains 50 red sweets and 50 blue sweets. I take 12 sweets from the bag.
X = The number of red sweets
4) I catch the bus top school every day. IN a month I catch the Bus 25 times.
X = The number of times that the bus is late
5) I flip a fair coin 20 times. X = The number of throws until I get my first head.
6) I watch my favourite football team play eight games. X = The number of games
that they win.
7) A computer has 5 components. If more than 3 of these components fail the
computer will not start. I switch on 50,000 computers. X = The number of computers
that fail to start.
Look at the graph of B(10,0.3).
1) Look at each of the Random Variables below. Fill in the gaps in each description:
I roll a red die and a blue die 8 times
Let X = The number of sixes rolled on the red die
We say that X ~ B(…,....)
Let Y = Number of evens scored on the blue die
We say that Y ~ B(…,....)
Let Z = The Number of times the blue die and red die add up to a score over 10
We say that Z ~ B(…,....)
a) Find P(X = 2)
b) Find P(Y = 3)
c) Find P(Z = 2)
d) Find P(X ≤ 2)
e) Find P(Y < 2)
f) Find P(Z ≥ 2)
2) Let X~B(10,0.25)
Find a) P(X = 2)
b) P(X = 0)
c) P(X<4)
d) P(X>8)
Lesson Objective
Be able to answer exam style questions involving Binomial situations.
Begin to use probability tables to calculate cumulative Binomial probabilities
Suppose X is a Random Variable
X ~ B(8,0.2)
a) Describe a situation in real life that could be modelled using this distribution.
b) Calculate P(X = 2)
c) Calculate P(X ≤ 2)
d) Calculate P(X ≤ 8)
1) The probability that a pen, selected at random from a production line of pens is
defective is 0.1. If a sample of 6 pend is taken. Find:
a) Probability that the sample contains no defective pens.
b) Probability that it contains 5 or 6 defective pens.
c) Probability that it contains less than 3 defective pens.
2) Assuming that boys and girls have an equal chance of being born. Find the
probability that in a family of 5 children there are more boys than girls.
3) The probability that a shopper chooses Soapysuds when buying washing
powder is 0.65. Find the probability that in a sample of 8 shoppers, the number
who choose Soapysuds is:
a) exactly 3
b) more than 5
4) 1% of a box of a production line of light bulbs are faulty. What is the largest
sample size which can be taken if it is required that the probability that there
are no faulty bulbs in the sample is greater than 0.5?
5) If X ~ B(n,0.6) and P(X<1) = 0.0256 find n
6) The probability that a target is hit is 0.3. Find the least number of shots which
should be fired if the probability that the target is hit at least once is greater
than 0.95?
5) Extensive research has shown that 1 person in every 4 is allergic to a particular
grass seed. A group of 20 university students volunteer to try out a new treatment.
a) What is the expected number of allergic people in the group?
b) What is the probability that exactly two people in the group are allergic?
c) What is the probability that no more than two people in the group are allergic?
d) How large a sample would be needed for the probability of it containing at least
one allergic person to be greater than 99.9%.
6) A circuit board has 5 components. It will fail to work if at least 3 of the
components are faulty. If the probability of a faulty component is 3/8.
What is the probability that any given circuit board is will not work?
If you buy a box of 10 circuit boards. What is the likelihood that more
than 1 of the circuit boards in the box is faulty?
1) The probability that a pen, selected at random from a production line of pens is
defective is 0.1. If a sample of 6 pend is taken. Find:
a) Probability that the sample contains no defective pens.
b) Probability that it contains 5 or 6 defective pens.
c) Probability that it contains less than 3 defective pens.
2) Assuming that boys and girls have an equal chance of being born. Find the
probability that in a family of 5 children there are more boys than girls. 0.5
3) The probability that a shopper chooses Soapysuds when buying washing
powder is 0.65. Find the probability that in a sample of 8 shoppers, the number
who choose soapy suds is:
a) exactly 3
0.0808
b) more than 5
0.428
4) 1% of a box of a production line of light bulbs are faulty. What is the smallest
sample size which can be taken if it is required that the probability that there
are no faulty bulbs in the sample is greater than 0.5?
68
5) If X ~ B(n,0.6) and P(X<1) = 0.0256 find n
4
6) The probability that a target is hit is 0.3. Find the least number of shots which
should be fired if the probability that the target is hit at least once is greater
than 0.95?
9
7) Extensive research has shown that 1 person in every 4 is allergic to a particular
grass seed. A group of 20 university students volunteer to try out a new treatment.
a) What is the expected number of allergic people in the group?
b) What is the probability that exactly two people in the group are allergic?
c) What is the probability that no more than two people in the group are allergic?
d) How large a sample would be needed for the probability of it containing at least
one allergic person to be greater than 99.9%.
8) A circuit board has 5 components. It will fail to work if at least 3 of the
components are faulty. If the probability of a faulty component is 3/8.
What is the probability that any given circuit board is will not work?
If you buy a box of 10 circuit boards. What is the likelihood that more
than 1 of the circuit boards in the box is faulty?
Lesson Objective
Discover the formula for the expectation and variance of a Binomial distribution
Imagine you have a biased coin with Probability of getting a Head 1/3
Let X = Number of Heads from 3 coin tosses. X ~ B(3, 1/3)
Draw up a Probability Distribution table and calculate E(X) and Var(X) exactly
Imagine you have another biased coin with Probability of getting a Head ¼
Let Y = Number of Heads from 4 coin tosses. Y ~ B(4, ¼)
Draw up a Probability Distribution table and calculate E(Y) and Var(Y) exactly
Imagine you have another coin with Probability of getting a Head ½
Let Z = Number of Heads from 5 coin tosses. Z ~ B(5, ½)
Draw up a Probability Distribution table and calculate E(Z) and Var(Z) exactly
Imagine you have another coin with Probability of getting a Head 2/5
Let W = Number of Heads from 5 coin tosses. W ~ B(5, 2/5)
Draw up a Probability Distribution table and calculate E(W) and Var(W) exactly
What do you notice about the expectation and variance each time?
How does it relate to ‘n’ and ‘p’? Can you prove this?
In general if:
You have ‘n’ trials.
Each independent and each with a two outcomes, success and failure.
The probability of success ‘p’ remains fixed for each trial.
The Random Variable, X, counts the number of successes in ‘n’ trials.
Then X~B(n,p)
P(X=r) = nCr pr (1-p)n-r
E(X) = np
Var(X) = np(1-p)