Transcript Probability - mrsmartinmath

Probability and Chance
Random Experiment

An experiment is random if
– The outcome depends on chance (we are not
sure of the outcome (result))
– We can list all the possible outcomes in the
sample space
Example: Roll a die. We are not sure what
number will be face up but we know it will be
one of
= {1 , 2 , 3 , 4 , 5, 6}
Event

An event is a subset of the sample space

Ex : Event A is an even number
– A { 2, 4, 6}
Ex: Event B is a number less than 5
- B { 1, 2, 3, 4}
Set Theory
A ∩ B is A intersect
B
A ∪B is A union B
Compatible
Compatible events can occur at the
same time
 Incompatible events can not occur at
the same time





A and B can have a shared outcome
A∩B ( A intersect B) is {2, 4}
Event C is roll a 5 the C in incompatible with both A and B
A ∩ C is empty
B ∩ C is empty
Complementary Event

Events A and B are complementary if
together they make up the sample space and
there is nothing shared
AUB=
A∩ B=0
 Example: A roll an even number

B roll an odd number

Probability (P)

Probability is a measure of how likely it is
for an event to happen.
 We name a probability with a number from
0 to 1.
• If an event is certain to happen, then the probability of
the event is 1.
P=1
• If an event is certain not to happen, then the probability
of the event is 0.
P=0
Probability

If it is uncertain whether or not an event
will happen, then its probability is some
fraction between 0 and 1 (part ÷ whole).
 Part = # of possible favorable outcomes
 Whole = # of all possible outcomes
B
C
A
D
3
1
2
A
C B
1. What is the probability that the spinner
will stop on part A?
2. What is the probability that the
spinner will stop on
(a) An even number?
(b) An odd number?
3. What fraction names the
probability that the spinner will
stop in the area marked A?
Probability Questions

Lawrence is the captain of his track team.
The team is deciding on a color and all eight
members wrote their choice down on equal
size cards. If Lawrence picks one card at
random, what is the probability that he will
pick blue?
blue
blue
yellow
red
green
black
blue
black

A.
B.
C.
Donald is rolling a number cube labeled 1
to 6. Which of the following is LEAST
LIKELY?
an even number
an odd number
a number greater than 5
CHANCE
what are the odds?

Chance is how likely it is that something
will happen. To state a chance, we use a
percent or a ratio ( part : part)
0
½
1
Probability
Certain not
to happen
Equally likely to
happen or not to happen
Certain to
happen
Chance
0%
50 %
50:50
100%
Chance

When a meteorologist states that the chance
of rain is 50%, the meteorologist is saying
that it is equally likely to rain or not to rain.
If the chance of rain rises to 80%, it is more
likely to rain. If the chance drops to 20%,
then it may rain, but it probably will not
rain.
1 2
4 3
1. What is the chance of spinning a
number greater than 1?
4 1
2
3
5
2. What is the chance of spinning a
4?
3. What is the chance that the
spinner will stop on an odd
number?
4. What is the chance of rolling an even
number with one toss of on number cube?
Sample Spaces
A sample set refers to the complete set of all the possible outcomes
Example: Roll a die. What are all the possible outcomes?
Sample set “S”
S = {1,2,3,4,5,6}
Sample Spaces
Example: Toss a coin. What are all the possible outcomes?
Sample space “S”
S = {H, T}
Toss two coins. What is the sample space?
S = {HH, HT, TH, TT}
Events
A set of outcomes is referred to as an event.
A specific outcome (part of the whole)
For example, when rolling a die the outcomes
that are an even number would be
referred to as an event.
Event = {2,4,6}
S = {1,2,3,4,5,6}
It is clear that outcomes and events are
subsets of the sample space, S.
Events
A set of outcomes is referred to as an event.
A specific outcome (part of the whole)
For example, when rolling a die the outcomes
that are an even number would be
referred to as an event.
Event = {2,4,6}
S = {1,2,3,4,5,6}
It is clear that outcomes and events are
subsets of the sample space, S.
Sample Space versus Events
We use the symbol omega
instead of S
so that we don’t get mixed up with events
Events are given a capital letter
Ex
= {1, 2, 3, 4, 5, 6}
A = { 2, 4, 6}
The sample space is all the possible outcomes of
rolling a dice
The event A is rolling an even number.
Compound Events
Sometimes we are asked to find the probability
of one event OR another
Sometimes we are asked to find the probability
of one event AND another
What’s the difference?
Example:
What is the probability of rolling a 2 OR a 4?
Compound Problems: Multiple Events
What is the probability of rolling a 2 and a 4 if
two die are rolled?
S = {11, 12, 13,14,15,16,
21,22,23,24,25,26,
31,32,33,34,35,36,
41,42,43,44,45,46,
51,52,53,54,55,56,
61,62,63,64,65,66}
Event {2 and 4} = {24,42}
All possible outcomes = 36
Possible outcomes of the stated event = 2
Therefore the probability is 2 out of 36
P = 0.056
Compound Events
S = {1,2,3,4,5,6}
Event {2} or {4}
There are 2 possible outcomes out of 6
P = 2/6 P = 0.33
Logical connectors
And , Or
When we see the probability of event A and B we multiply
When we see the probability of event A or B, we add
Example: We roll a die, what is the probability of rolling
a 3 or a 5? 1/6 + 1/6 = 2/6 or 0.33
Example: We roll a die and then roll it again,
what is the probability of rolling a 3 and a 5?
1/6 x 1/6 = 1/36 (much less likely)
Compound Events
Independent versus Dependent
Events
Independent: if event A does not influence the probability
of event B
Dependent: if event A does influence the probability of
event B
Example: Event A: choose a marble
Event B: choose a marble
They are independent if I replace the marble, dependent if
I do not replace the marble
Compound Events
Independent versus Dependent
Events
Example: there are 100 skittles
20 red
20 orange
20 green
20 purple
20 yellow
What is the probability of choosing a red one, eating it
and then choosing a yellow one?
Are these events dependent or independent?
Compound Events
Independent versus Dependent
Events
Example: there are 100 skittles
20 red
20 orange
20 green
20 purple
20 yellow
What is the probability of choosing a red one, eating it
and then choosing a yellow one?
P(A) X P(B) = 20/100 X 20/99 (remember, I ate one)
Compound Events
Independent versus Dependent
Events
Example: there are 100 skittles
20 red
20 orange
20 green
20 purple
20 yellow
What is the probability of choosing 2 red one (I don’t
replace the first – obviously)
P(A) X P(B) = 20/100 X 19/99 (remember, I ate one)
Compound Events
Independent versus Dependent
Events
Example: there are 100 skittles
20 red
20 orange
20 green
20 purple
20 yellow
What is the probability of eating 1 orange, 1 green, 1 purple
and then 1 green?
20/100 x 20/99 x 20/98 x 19/97
Get it?
Compound Events
Independent versus Dependent
Events
Example: there are 100 skittles
20 red
20 orange
20 green
20 purple
20 yellow
Create your own question…