Name: _______________________
Class: __________
Math SL: 23C Expectation
Review:
6
1. Evaluate:
å3
k
k=2
2. Given f(x) = –3x2 – 6x – 7,
(a) rewrite f(x) in vertex form;
(b) determine the number of solution to f(x) = 0;
(c) determine all intercepts.
1
Date:_____________
23C Expectation
Today’s Objectives:
(1) to understand the meaning of Expectation
(2) to be able to calculate the expectation of a random variable
Expectation
The expected value (mean) of a random variable is the average value that we should
expect for X over many trials of the experiment. We can also talk about the expected
outcome from one trial of an experiment. The symbol for expected value is E(X) or m .
If there are n trials of an experiment, and an event has a probability p of occurring in
each of the trials, then the number of times we expect the event to occur is np.
In general, the expectation of the random variable X is:
n
n
E(X) = m = å xi pi
or
i =1
å x P(X = x )
i
i
i =1
Example 1: When throwing a single standard six-sided dice, let S = the square of the
score shown on the dice. What is the expectation of S?
Example 2: The random variable Z has the below probability distribution and E(Z) = 5 23 .
Find x and y.
z
2
3
5
7
11
1
1
1
P(Z = z)
x
y
6
6
6
2
Fair Games
In gambling, we say that for each game:
expected gain of the player = expected return or payoff – the amount it cost them to
play
The game will be fair if the expected gain is zero. Suppose X represents the gain of a
player from each game.
The game is fair if E(X)=0.
Example 3: In a game of chance, a player spins a square spinner labeled 1, 2, 3, 4. The
player wins the amount of money shown in the table below, depending on which
number comes up.
Number
Winnings
1
$1
2
$2
3
$5
4
$8
Determine:
a) The expected return for one spin of the spinner.
b) The expected gain of the player if it cost $5 to play each game.
c) Whether this is a fair game or not.
Example 4: If two dice are rolled simultaneously 180 times, on how many occasions
would you expect to get a double?
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Example 5:
Determine the expected IB Math SL grade
X=Math SL Grade
P( X = x)
1
2
3
4
5
6
7
1.4%
10.1%
17.6%
19.3%
23.8%
19.1%
8.7%
Is this an accurate estimate of how you will perform in May 2015? Explain.
Hmwk#52
23C Expectation pg. 637 # 2 – 14 even
Note: I would strongly recommend that you start to review for the next test. If you have
any questions regarding topics/concepts, be proactive and seek help. If you feel you need
more practice, then be proactive and COMPLETE more problems.
Continue researching a math exploration topic – find math involved.
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