Download File - DP Mathematics SL

PROBABILITY DISTRIBUTIONS
DISCRETE RANDOM VARIABLES
OUTCOMES & EVENTS
Mrs. Aldous & Mr. Thauvette
IB DP SL Mathematics
You should be able to…
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Construct a discrete probability distribution
Calculate expected value from a discrete
probability distribution
Apply the concept of expectation to a variety of
situations, including games of chance.
Challenge
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How does rolling a die and doubling the score
differ from rolling two dice and adding the scores?
Random Variables
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A random variable is a rule that assigns exactly one
value to each point in a sample space for an
experiment.
A random variable can be classified as being either
discrete or continuous depending on the numerical
values it assumes.
A discrete random variable may assume either a finite
number of values or an infinite sequence of values.
A continuous random variable may assume any
numerical value in an interval or collection of intervals.
Random Variables
Question
Family
size
Random Variable x
x = Number of dependents in
Discrete
family reported on tax return
Distance from
x = Distance in miles from
home to store
home to the store site
Own dog
x = 1 if own no pet;
or cat
Type
= 2 if own dog(s) only;
= 3 if own cat(s) only;
= 4 if own dog(s) and cat(s)
Continuous
Discrete
Probability Distributions
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The probability distribution for a random
variable describes how probabilities are
distributed over the values of the random
variable.
For example:
Probabilities of flipping x heads from two coin tosses
X – is the random variable for the event
‘number of heads’
x – is the number of heads for the
calculations
Number of heads
(x)
P(X=x)
Probability of the
event X being ‘x’
0
1/
4
1
1/
1/
+
4
4
= 1/2
2
1/
4
Who wants some money?
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Here are the rules. You throw a die and if you throw
n, I will give you $n thousand.
Would you play?
Would you pay to play? Why?
If so, what is the maximum you would pay to play?
How much would you expect to make in the long
term if you kept playing this game with me?
Expectation

The mean of the random variable X is called the
expected value of X
…it is written E(X)
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The expected value of X is:
E( X )   x P( X  x)
Expectation: Simple Example
The expected value of X is
Number of heads
(x)
0
1/
P(X=x)
Probability of the
event X being ‘x’
E( X )   x P( X  x)
4
1
1/
1/
+
4
4
2
1/
4
= 1/2
E(X) = 0 x 1/4 + 1 x 1/2 + 2 x 1/4
= 1 “You would expect 1 head out of every 2
throws”
Now you try
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Sasha’s pocket contains one $1 coin, one 50 c coin,
and three 20 c coins. He selects two coins at random
to place in a charity collection box. The random
variable X represents the amount, in cents, that he
puts in the box.
(a) Show that P(X = 70) = 0.3.
(b) Find the probability distribution for X.
(c) Find the expected value of X.
Probability distribution tables
100 professional basketball players are surveyed. Each player has 5
3-point throws at a basket and the number of baskets scored is
recorded in the table below.
Baskets
Frequency
0
10
1
10
2
25
3
35
4
10
5
10
Using this table can you find the probability of a basketball player
being picked at random scoring,
a) 1 basket,
10
= 0.1
100
b) 2 basket,
25
= 0.25
100
c) 3 baskets,
35
= 0.35
100
d) 0 baskets?
10
= 0.1
100
Turn this into a probability
distribution table...
Probability tables
Baskets
Frequency
0
10
1
10
2
25
3
35
4
10
5
10
Baskets
Probability
0
0.1
1
0.1
2
0.25
3
0.35
4
0.1
5
0.1
From the table of probabilities we can find the average expected
number of baskets:
(0 ´ 0.1) +(1´ 0.1) + (2 ´ 0.25) + (3´ 0.35) + (4 ´ 0.1) + (5 ´ 0.1)
= 2.55
This is called the
expected value, or E(x).
You can also use a GDC, List 1: the baskets, List 2: the frequencies.
Another example
The random variable X has a probability function given by the
formula,
3x - 2
for x = 2,3,4,5,6.
50
a) Complete the table below to show the probabilities of x.
x
2
3
4
P(X=x)
4
50
7
50
10
50
substitute 2 into
the formula:
3(2) - 2 4
=
50
50
5
13
50
6
16
50
now complete the table.
b) Use your table to find the probability that x>4.
c) Find E(x), the expected outcome of x.
E(X) =
E(X) =
13 + 16 29
=
50
50
8 + 21+ 40 + 65 + 96
50
230
50
Question
1. The table below shows the probabilities of a random function, x.
x
1
2
3
4
5
6
Pr(X=x)
0.2
0.25
p
0.05
0.1
0.1
a) Find the value of p.
b) Hence find E(x).
p=0.3
E(x)=2.9
Probability Distributions
Here are 2 of the probability distributions from the
questions you have done so far. What do you notice?
Number of heads
(x)
0
1/
P(X=x)
1
1/
4
Probability of the
event X being ‘x’
1/
+
4
4
2
1/
4
= 1/2
x
1
2
3
4
5
6
P(X=x)
0.2
0.25
0.3
0.05
0.1
0.1
Make your own probability distribution
Place all the numbers from the bag
into the table below (with no
repeats!) to make a viable
probability distribution.
How many can you make?
Question
A die is biased such that the probabilities of the die
landing on each score is given below:
(a) Show that 5x + 10y = 0.9
(b) Given that E(S) = 3.35, find the values of x and y.
Exam Style Question
Two fair four-sided dice, one red and one green, are
thrown. For each die, the faces are labelled 1, 2, 3, 4.
The score for each die is the number that lands face
down.
(a) Write down
(i) the sample space
(ii) the probability that two scores of 4 are
obtained
Exam Style Question Continued…
(a) Write down
(i) the sample space
(a) Write down
(ii) the probability that two scores of 4 are
obtained
Exam Style Question Continued…
Let X be the number of 4s that land face down.
(b) Copy and complete the following probability
distribution for X.
Let X be the number of 4s that land face down.
Exam Style Question Continued…
(c) Find E(X).
You should know…
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A random variable, denoted by X, takes on
observed values x1, x2, x3, …, xn, and if the data
obtained can be counted, then X is a discrete
random variable
The probability that the random variable X can
take on the values xi is denoted by P(X = xi)
A discrete probability distribution lists the values of
the random variable with their corresponding
probabilities
You should know…
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A probability distribution may be presented in
tabular form or as a function. For example,
1
P ( X = x ) = ( 4 + x ) for x = 0, 1, 2, 3
22
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The sum of all probabilities in a distribution is 1
You should know…
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The expected value of a distribution is its mean and
is denoted by E(X) or m . It is determined by
summing the product of the outcomes with their
corresponding probabilities, that is,
E(X) = m = å ( xP ( X = x ))
x
Be prepared…
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Although expected value is an average, there is no
need to divide by n as the formula for expected
value already takes this into account.