Download Math 1313 Section 6.1 experiment

Math 1313
Section 6.1
Section 6.1: Experiments, Events, and Sample Spaces
An experiment is an activity with observable results (outcomes).
A sample point is an outcome of an experiment.
A sample space is a set consisting of all possible sample points of an experiment.
A Finite Sample Space is a sample space with finitely many outcomes.
An event is a subset of a sample space of an experiment.
Given two events, E and F:
The union of E and F is denoted by ∪ .
The intersection of E and F is denoted by ∩ .
If ∩ = Ø then E and F are called mutually exclusive. (An event is mutually exclusive also
means that two events that cannot happen at the same time, such as getting a head and a tail on
the same toss of a coin).
The complement of an event is and is the set of all outcomes in a sample space that is not in E.
Example 1: Consider the experiment of tossing a die.
a. Describe the sample space, S, of this experiment.
b. Let E be the event that an even number is tossed and F be the event that a prime number is
tossed. Describe E and F in set notation then find the following:
∪ =
∩ =
=
∪ =
1
Math 1313
Section 6.1
Example 2: A sample of 3 apples taken from a fruit stand is examined to determine whether they
are good or rotten. The sample space S = {GGG, GGR, GRG, GRR, RGG, RGR, RRG, RRR}. Let E be the
event that at least 1 apple is good and let F be the event that exactly 2 apples are rotten. Find the
events.
Example 3: An experiment consists of selecting a letter at random from the letters in the word
DALLAS and observing the outcomes.
a. What is an appropriate sample space for this experiment?
b. Describe the event “the letter selected is a vowel.”
Example 4: Describe a sample space associated with the experiment of tossing 2 fair coins.
Describe the event of having the same outcome on each coin.
Popper 1: Given the following sample space, describe the event of selecting a prime number.
= {1, 2, 3, 4, 6, 8, 9, 11, 12, 15, 16, 18, 19, 20}
a.
b.
c.
d.
e.
E = {2, 4, 8, 16}
E = {1, 2, 3, 9, 11, 19}
E = {2, 3, 11, 19}
E = {1, 3, 9, 11, 15, 19}
None of the above
2
Math 1313 Section 6.2
Section 6.2: Introduction to Probability
The ratio
௠
௡
is the relative frequency of an event E that occurs m times after n repetitions.
Note: The probability of an event is a number that lies between 0 and 1, inclusive.
If S={s1, s2 ,…, sn } is a finite sample space with n outcomes, then the events {s1}, {s2},…, {sn} are
called simple events of the experiment.
Once probabilities are assigned to each of these simple events, we obtain a probability distribution.
The probabilities, P(s1), P(s2),…, P(sn) have the following properties:
1. 0 ≤ Pሺs୧ ሻ ≤ 1, i = ሼ1, 2, 3, … , nሽ
2. Pሺsଵ ሻ + Pሺsଶ ሻ + ⋯ + Pሺs୬ ሻ = 1
3. P൫s௜ ∪ s௝ ൯ = Pሺs௜ ሻ + P൫s௝ ൯, ݅ ≠ ݆ and ݅, ݆ = 1,2,3, … n
Example 1: A fair die is cast. List the simple events.
A sample space in which the outcomes of an experiment are equally likely to occur is called a uniform
sample space.
Let S={s1, s2,…, sn} be a uniform sample space. Then
ܲሺsଵ ሻ = ܲሺsଶ ሻ = ⋯ = ܲሺs௡ ሻ =
1
݊
Finding the probability of an Event E:
1. Determine the sample space S.
2. Assign probabilities to each of the simple events of S.
3. If E={ s1, s2,…, sk} where {s1}, {s2},…, {sk} are simple events then
ܲሺ‫ ܧ‬ሻ = ܲሺ‫ݏ‬ଵ ሻ + ܲሺsଶ ሻ + ⋯ + ܲሺs௞ ሻ
Note: If E = Ø then P(E) = 0.
1
Math 1313 Section 6.2
Example 2: The accompanying data were obtained from a survey of Americans who were asked: How
safe are American-made consumer products
Rating
Very Safe
Somewhat safe
Not too safe
Not safe at all
Don't know
Number of Respondents
76
244
60
8
12
Find the probability distribution associated with this experiment.
Example 3: A pair of fair dice is cast. What is the probability that
a. the sum of the numbers shown is less than 5?
b. at least one 6 is cast?
c. you roll doubles?
2
Math 1313 Section 6.2
Example 4: If one card is drawn from a well-shuffled standard 52-card deck, what is the probability that
the card drawn is
a. A club?
b. A red card?
c. A seven?
d. A face card?
e. A black 9?
3
Math 1313 Section 6.2
Popper 3: A pair of fair dice is cast. What is the probability that the sum of the numbers falling
uppermost is 7?
a. 0.5833
b. 0.1667
c. 0.1944
d. None of the above
Example 5: A survey was taken in a certain community about the number of the radios in the house, the
probability distribution was constructed:
Number of Radios
Probability
0
1
2
3
0.01
0.09
0.53
0.37
What is the probability of a house chosen at random from this community having,
a. 1 or 2 radios?
b. more than 1 radio?
c. not even one radio?
Popper 2: If one card is drawn from a well-shuffled standard 52-card deck, what is the probability that
the card drawn is a red six?
a.
b.
c.
d.
e.
1/52
1/26
1/13
2/13
None of the above
4
Math 1313
Section 6.3
Section 6.3: Rules of Probability
Let S be a sample space, E and F are events of the experiment then,
1. P ≥ 0, for any event E.
2. P = 1
3. If E and F are mutually exclusive, ∩ = Ø, then ∪ = + .
(Note that this property can be extended to a finite number of events.)
4. If E and F are not mutually exclusive, ∩ ≠ Ø, then
∪ = + – ∩ (Note that this property can be extended to a finite number of events.)
5. (Rule of Complements) If E is an event and Ec denotes the complement of E then
= 1 – .
Example 1: An experiment consists of selecting a card at random from a well-shuffled deck of 52
playing cards. Find the probability that an ace or a spade is drawn.
Example 2: Let E and F be two events and suppose that = 0.37, = 0.3 and
P ∩ = 0.08. Compute
a. ∪ b. c. ∩ 1
Math 1313
Section 6.3
Popper 4: Let E and F be two events and suppose that = 0.43, = 0.38 and
P ∩ = 0.08. Find P(EC)
a.
b.
c.
d.
0
0.43
0.62
0.57
Example 2: The SAT Math scores of a senior class at a high school are shown in the table.
Range of
Scores
> 700
# of Students in the
Range
16
600 < ≤ 700
94
500 < ≤ 600
165
400 < ≤ 500
309
300 < ≤ 400
96
< 300
20
a. Construct the probability distribution for the data.
b. If a student is selected at random, what is the probability that his or her score was:
i. More than 500?
ii.
Less than or equal to 400?
iii.
Greater the 400 but less than or equal 700?
2