Download 1.4 Basics of Probability

1.4 Basics of Probability
MATH 166-506, Fall 2016
Jean Yeh
1.4 Basics of Probability
Definition: A uniform smaple space is a sample space whose individual elementary events are
equally likely.
Example: The smaple space for the flipping coins experiment, the rolling dice experiment, the
drawing cards experiment are all uniform sample space.
Note: Normally, we will consider a fair coin, a fair die, or drawing a card randomly.
Definition: If S is a finite uniform sample space and E is any event, then the probability of E,
P(E) is given by
Number of elements in E
n(E)
p(E) =
=
Number of elements in S
n(S)
1. P (E) = 0, if E = ∅.
Note: For any event E, we have 0 ≤ P (E) ≤ 1. We also have:
2. P (E) = 1, if E = S
The larger the P (E) is, the higher chance that event E will occur.
Example: Suppose a single card is randomly drawn from a standard deck of 52 playing cards.
• What is the number of sample space for this experiment?
The sample space is S = {♠1 ∼ ♠13, ♥1 ∼ ♥13, ♦1 ∼ ♦13, ♣1 ∼ ♣13. Hence, n(S) = 52
Determine the probability of each of the following events.
• The event E: ”Drawing a 5”.
The event E = {♠5, ♥5, ♦5, ♣5}, therefore n(E) = 4. We have
n(E)
4
=
n(S)
52
• The event F : ”Drawing a diamond”.
The event F = {♦1 ∼ ♦13} , therefore n(F ) = 13. We have
n(F )
13
=
n(S)
52
• The event G: ”Drawing a black card”.
The event G = {♠1 ∼ ♠13, ♣1 ∼ ♣13}, therefore n(G) = 26. We have
n(G)
26
=
n(S)
52
• The event H = (E ∪ F ∪ G)c .
The event H = (E ∪ F ∪ G)c = E c ∩ F c ∩ Gc = {♥1 ∼ ♥4, ♥6 ∼ ♥13} , therefore
n(H)
12
n(H) = 12. We have
=
n(S)
52
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1.4 Basics of Probability
MATH 166-506, Fall 2016
Jean Yeh
Definition: The empirical probability of an event is a practical estimate probability for the event
which calculated based on the real data.
After repeat the experiment M times, the event E occured N times. We said the empirical probability of event E is
N
P (E) =
M
Example: A candy company survey about the likelihood of a new type of candy. The data recorded
below: (total 169 surveyees)
age
6∼12 6∼12 13∼21 13∼21 22∼30 22∼30 31∼40 31∼40 41up 41up
likelihood
y
n
y
n
y
n
y
n
y
n
surveyee
23
3
30
8
27
13
22
16
13
14
• What’s the empirical probability of ”survive to age bewteen 31 and 40, also like the new
type of candy”?
The number of surveyee is 22, therefore, the empirical probability is 22/169.
• What’s the empirical probability of ”survive to age bewteen 6 and 12”?
The number of surveyee is 23+3=26, therefore, the empirical probability is 26/169.
Definition: A probability distribution table is a table to display probability data for an experiment.
The event chosen must be mutually excusive and therefore the total probability will add to 1.
Example: Two fair dice are rolled. Find the probability distribution table for the sum of the
numbers shown uppermost.
We will find the number of all the possible events.
sum
2 3 4 5 6 7 8 9 10 11 12
number of event 1 2 3 4 5 6 5 4 3 2 1
total:36
Therefore, we can got the probability distribution table
sum
2
3
4
5
6
7
8
9
10
11
12
probability 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
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