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Chance Variables
1. Chance variables.
When a pair of dice are rolled, the total number of points that show
could be either 2, 3, … 11, or 12. When a coin is tossed 10 times, the
number of times a head comes up could be 0, 1, … 9, or 10. The total
number of points, and the number of times a head comes up, are
examples of
chance variables
. In the Frst example, the numbers 2
through 12 are said to be the
possible values
of the total number of
points. In the second example, the numbers 0 through 10 are the pos
sible values of the number of times heads comes up. People often drop
the “possible”, and just use “the values”.
In mathematics, entities that can have different values within the
same context are often referred to as variables. The total number
of points (when a pair of dice are rolled) is a variable because it is
possible for it to be any number between 2 and 12 inclusive. It is a
“chance” variable because the possible values have chances: the value
2 has chance 1/36, the value 3 has chance 2/36, and so on.
Example 1.
If a roulette player stakes $1 on red at roulette, two things can hap
pen. Either he wins, in which case the casino returns his stake along
with another dollar as prize; or he loses, in which case the casino
keeps the stake.
A roulette player bets $1 on red six times in a row. His
net gain
from
the six bets is deFned as:
net gain = amount he wins — amount he loses
Net gain is a chance variable. What are its possible values?
Answer.
What is the best that could happen to the gambler? He could win ev
ery time. In that case his net gain would 6 (dollars). What is the worst
that could happen? He could lose every time, and the net gain would
be –6 (dollars). So the possible values are whole numbers between –6
and 6 (inclusive). But not all whole numbers. Suppose the gambler
wins Fve times and loses once. His net gain would be 4 (dollars). The
possible values of net gain are: –6, –4, –2, 0, 2, 4, 6.
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2. Distribution table of a chance variable.
The distribution table of a chance variable is a table with two col
umns. The Frst column lists the possible values of the variable. The
second column displays, beside each value in the Frst column, the
chance the variable turns out to be that value.
Example 2.
A die is rolled twice. The number of times the single dot (
)shows
up in the two rolls is a chance variable. ±ind its distribution table.
Answer.
The Frst step is to get the values of the chance variable. The single
dot could show up on neither roll, one of the rolls, or both. So the pos
sible values are 0, 1, 2. That gives the Frst column of the distribution
table:
value
0
1
2
To get the second column, you have to work out the three chances
below:
value
0
chance the single dot shows up 0 times
1
chance the single dot shows up 1 time
2
chance the single dot shows up 2 times
That leads to the following table:
value
chance
0
25/36
1
10/36
2
1/36
This is the distribution table for the number of times
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 Fall '08
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