IB Math SL Year 2
Name: ____________________________
3-3 Probability Distribution and Expected Value
Classwork
In this lesson, we will answer the following learning goals:
1. What is a random variable and how do you list its probability outcomes?
2. How do you predict the average value of a random variable?
1. Probability Distribution
A __________________ is a quantity whose value depends on chance, for example, the outcome when a die is
rolled.
A _____________ random variable X has possible
A ____________________ random variable X could
take possible values in some interval on the number
values
line.
x1, x2, x3, …
Determined by __________________
Determined by ____________________
Example
Example
The heights of men could all lie in an interval 50 <x<250
The number of bicycles sold each year by a bike store
cm
Recall that the sample space of a random event is the list of all possible outcomes of the event
Key Idea: Probability Distribution
The sample space of a random variable, with the probabilities associated with all of the values in sample space
Example:
Key Points to Consider
1. The sum total of all the probabilities in a probability distribution is always equal to 1
2. All probabilities in a probability distribution must be 0<x<1
3. Probability distributions of a discrete random variable can be given in a table, graph or probability mass
function P(x).
Let’s try it.
Example 1) Find k in the following probability distributions:
IB Math SL Year 2
Application Problems
Example 2) Bryana’s number of hits in each softball match has
the following probability distribution:
a) State clearly what the random variables represent.
b) Determine and state the value of k
c) Determine:
i. P(3)
ii) P(X≥ 2)
iii) P(1≤ 𝑥 ≤ 3)
Example 3) A die is rolled twice and the sum of the faces is recorded.
a) Draw a grid which displays the sample space
b) Tabulate a probability distribution of X if X denotes the sum of the results for the two rolls.
c) Determine:
i) P(x=3)
ii) P(D≥ 8|𝐷 ≥ 6)
Example 4) The probability distribution of a random variable Y is given by P(Y=y) = cy3 for y = 1,2,3
Given that c is a constant, find the value of c.
Y
cy3
P(Y=y)
IB Math SL Year 2
2. Expected Value of a discrete random variable
The ____________________ of a random variable is the ___________ value you would get if you were to repeatedly
measure the variable an infinite number of times
The mean or expected value of a random variable 𝑋 is represented by 𝐸(𝑋)𝑜𝑟 𝜇 and is defined by the following
formula:
Let’s see how it works:
Example 1) When throwing a standard six-sided dice, let X be the random variable defined by X = the square of the
score shown on the dice. Find the exact value of E(X).
X
P(X=x)
1
4
9
16
25
36
Example 2) The random variable V has the probability distribution as shown in the table and E(V) = 6.3. Find the value of
k.
V
E(V)
1
0.2
2
0.3
5
0.1
8
0.1
K
0.3
Example 3) A random variable X has its probability distribution given by P(X =x) = k(x+3) where x is 0,1,2, or 3.
1
a) Show that k = 18.
b) Find the exact value of E(X).
IB Math SL Year 2
Still have time? Mixed Practice
1. The random variable X has the following distribution. Find k.
x
P(X=x)
-1
2k
0
4𝑘 2
1
6𝑘 2
2
k
2. The probabilities of Nick scoring home runs in each game during his baseball career are given in the following table. X
is the number of home runs per game.
a) What is the value of P(2)?
b) What is the value of a? Explain what this number means in the context of this problem.
𝑘
3. Find the value k for the probability distribution described as: P(X=x) = 𝑥 for x = 1,2,3,4
4. Calculate the expectation of each of the following random variables:
x
P(X=x)
8
0.4
9
0.3
10
0.2
11
0.1