Combinatorics
Probability & Statistics
© The scientific sentence. 2010
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Probabilities
1. ExampleLet's consider again the example of throwing two
six-sided dice. From our previous results, we can
set the following tables. In the first table, we
find the 36 possibilities of the sum of the two
outcome numbers. In the second table, we set the
probability for each possible sum. For example,
the sum equal to 5 occurs with 4 outcomes: (1,4), (2,3),
(3,2), and (4,1). The total number of all possible
outcomes is 36.
Table 1
first
die |
second die |
| 1 |
2 |
3 |
4 |
5 |
6 |
| 1 |
2 |
3 |
4 |
5 |
6 |
7 |
| 2 |
3 |
4 |
5 |
6 |
7 |
8 |
| 3 |
4 |
5 |
6 |
7 |
8 |
9 |
| 4 |
5 |
6 |
7 |
8 |
9 |
10 |
| 5 |
6 |
7 |
8 |
9 |
10 |
11 |
| 6 |
7 |
8 |
9 |
10 |
11 |
12 |
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Table 2
| sum (event) |
number of related
outcomes |
probability |
| 2 |
1 |
1/36 |
| 3 |
2 |
2/36 |
| 4 |
3 |
3/36 |
| 5 |
4 |
4/36 |
| 6 |
5 |
5/36 |
| 7 |
6 |
6/36 |
| 8 |
5 |
5/36 |
| 9 |
4 |
4/36 |
| 10 |
3 |
3/36 |
| 11 |
2 |
2/36 |
| 12 |
1 |
1/36 |
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2. Probabilities
2.1. Theoritical probabilityIf we consider the event "the sum of the outcome numbers is equal to 5",
we write E = {(1,4),(2,3),(3,2),(4,1)}, and |E| = 4. The oucomes
in the event E are equally likely. With |S|= 36, the probability
to have a sum equal to 5 when rolling two fair six-sided dice is 4/36 = 1/9.
This theoritical probability can be generalized to:
The brobability P(E) that an event E occurs among
S possibilities is equal to |E|/|S|
P(E) = |E|/|S|
The probability that an outcome will occur =
number of favorable outcomes /number of possible outcomes
2.2. Empirical probability
If we repeat the random experiment of throwing the two
dice, N times, and record the number NE of times each
given event E (sum) occurs, the rate NE/N will be close
to the theoritical probability P(E) defined above.
The number NE is called frequency. NE/N is the relative frequency.
The more the number of repeatition N is big, the more the
empirical probability becomes close to theoritical probability.
We write this as:
P(E) = lim (NE/N)
N → ∞
This is the Jacob Bernouilli's theorem or the law of large numbers.
2.3. Other form: probability by measurement
The sample space S can be finite, infinite, discrete,
or continuous. The sample space S = {1,2,3,4,5,6} for rolling a
six-sided dice is finite and discrete. We can assign a probability
for each event from S. We cannot do this when S is continuous.
In this case, we assign a probability related to a measure done
inside the sample space.
Example:
If we throw an arrow towards a circle of radius R = 30 cm,
how likely is it that the arrow will land at a distance not far
from r = 10 cm from the center of the circle?
We calculate the related prbability by measure:
Here the sample space is continuous, S = πR2,
and E = πr2; (the event E is "to land in the circle
of radius r"). We have:
P(E) = E/S = πr2/πR2 = (r/R)2
= (10/30)2 = 1/9.
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