Transcript Probability - math

Probability
www.mathsrevision.com
S4 General
Likelihood Number Line
Combined Probabilities
Relative Frequency
Mutually Excusive Events
Independent Events
29-Mar-16
Starter Questions
www.mathsrevision.com
S4 General
1.
Find the value of x if
2x-3=17
2.
Calculate (-5)+12 - 15
3.
Calculate x:
4.
Change 790cm into m
5.
If I toss a coin in the air
what is the chance of getting a head?
29-Mar-16
x
o
75
o
Probability
www.mathsrevision.com
S4 General
Learning Intention
1. To understand probability
in terms of the number
line and calculate simple
probabilities.
Success Criteria
1. Understand the probability
line.
2. Calculate simply probabilities.
29-Mar-16
Probability
Likelihood Line
www.mathsrevision.com
S4 General
0
Impossible
Seeing
a butterfly
In July
29-Mar-16
0.5
Not very
likely
School
Holidays
Evens
Winning the
Lottery
1
Very
likely
Baby Born
A Boy
Certain
Go back
in time
Probability
Likelihood Line
www.mathsrevision.com
S4 General
0
Impossible
It will
Snow in winter
29-Mar-16
0.5
Not very
likely
Evens
Homework Everyone getting
Every week
100 % in test
1
Very
likely
Certain
Toss a coin Going without
That land
Food
Heads
for a year.
Probability
www.mathsrevision.com
S4 General
Key Points
To work out a probability
number of outcomes
P(A) =
Total number of possible outcomes
Probability is ALWAYS in the range 0 to 1
29-Mar-16
Probability
Number Likelihood Line
S4 General
www.mathsrevision.com
1
0
2
0.1
Impossible
3
0.2
4
0.3
5
0.4
6
0.5
0.6
8
7
0.7
0.8
0.9
Evens
Q. What is the chance of picking a number between 1 – 8 ?
1
Certain
P=
8
=1
8
4
P(E) =
= 0.5
8
Q. What is the chance of picking the number 1 ?
1
P(1)
=
= 0.125
29-Mar-16
8
Q. What is the chance of picking a number that is even ?
Probability
Likelihood Line
S4 General
52 cards in a pack of cards
0
0.1
Impossible
0.2
0.3
Not very
likely
0.4
0.5
0.6
0.7
Evens
0.8
Very
likely
0.9
Q. What is the chance of picking a red card ?
26
P (Red) =
52
Q. What is the chance of picking a diamond ?
P (D) =
Q. What is the chance of picking ace ?
4
52
29-Mar-16
P (Ace) =
1
Certain
= 0.5
13
= 0.25
52
= 0.08
Probability
www.mathsrevision.com
S4 General
Now try Ex 1A & 1B
Ch10 (page 123)
29-Mar-16
Starter Questions
S4 General
www.mathsrevision.com
1.
Find the value of x if
3x-3=27
x
o
2.
Calculate x:
3.
What is the chance of picking a face
card from a pack of cards.
1
4. 33 % of 240
3
29-Mar-16
37
o
Probability
Combined Probabilities
S4 General
Learning Intention
1. To understand combined
probability by constructing
tables or tree diagrams.
29-Mar-16
Success Criteria
1. Construct combination tables
and tree diagrams.
2. Calculate probabilities from
tables or tree diagrams.
Traffic Light 1
Probability
Combined Probabilities
S4 General
www.mathsrevision.com
Traffic Light 2
On route to school I pass two sets of traffic light.
They work independently.
Complete the table of all possible combinations
Traffic light 2
R
A
G
r (r,R)
(r,A)
(r,G)
Traffic a (a,R)
Light 1
g (g,R)
(a,A)
(a,G)
(g,A)
(g,G)
29-Mar-16
Calculate Probability :
1. Both green =
1
9
2. At least one green =
5
9
3. One is amber one red
2
=
9
Traffic Light 1
Probability
Combined Probabilities
S4 General
www.mathsrevision.com
Traffic Light 2
Another way of representing the data !
Tree Diagram
(r,R)
R
Calculate Probability :
A
T L2
(r,A)
1
G
r
(r,G)
1. Both green =
TL1
R
a
T L2 G
g
(g,R)
A (g,A)
(g,G)
G
R
T L2
29-Mar-16
A
(a,R)
(a,A)
(a,G)
9
2. At least one green =
5
9
3. One is amber one red =
2
9
Probability
Combined Probabilities
S4 General
www.mathsrevision.com
Key Points
1.
List all combinations
(using a suitable table or tree diagram)
2.
Interpret information to calculate probabilities.
29-Mar-16
Probability
Combined Probabilities
www.mathsrevision.com
S4 General
Now try Exercise 2
Ch10 (page 124)
29-Mar-16
Starter Questions
www.mathsrevision.com
S4 General
1.
Find the value of x if
2x-5=9
2.
Calculate x o :
2 4
3.
+
3 5
4.
29-Mar-16
o
120
x
o
What the the probability of throwing
a double six with a pair of dice.
Probability
Relative Frequency
www.mathsrevision.com
S4 General
Learning Intention
1. To understand the term
relative frequency.
Success Criteria
1. Know the term relative
frequency.
2. Calculate relative frequency
from data given.
29-Mar-16
Probability
Relative Frequency
S4 General
www.mathsrevision.com
Relative Frequency :
Example
How often an event happens
compared to the total number
of events.
Daniel
Frances
Number of heads
24
55
Number of tosses
60
100
24 2
=
= 0.4
Joe has a R.F. of heads =
60 5
29-Mar-16
Probability
Relative Frequency
S4 General
www.mathsrevision.com
Relative Frequency :
Example
How often an event happens
compared to the total number
of events.
Daniel
Frances
Number of heads
24
55
Number of tosses
60
100
55
= 0.55
Frances has a R.F. of heads =
100
29-Mar-16
Probability
Relative Frequency
www.mathsrevision.com
S4 General
Joe has a R.F. of heads = 0.4
Frances has a R.F. of heads = 0.55
Whose relative frequency is closer to P(H) ?
P(H) = 0.5
France’s R.F. is closer to P(H)
Note : The larger the random sample, the closer
the relative frequency is to the probability of the event
29-Mar-16
Probability
Relative Frequency
www.mathsrevision.com
S4 General
Now try Exercise 3
Ch10 (page 126)
29-Mar-16
Starter Questions
S4 General
www.mathsrevision.com
1.
Find the value of x if
4x =48
o
xo :
47
2.
Calculate
3.
What is the probability of picking
a number less than 4 from a pack of cards
4.
Find the average of the numbers 15, 25, 20?
29-Mar-16
o
55
x
o
Probability
Mutually Exclusive
www.mathsrevision.com
S4 General
Learning Intention
1. To understand that the
term mutually exclusive.
Understand that we simply
add probabilities with this
property.
29-Mar-16
Success Criteria
1. Understand the term
mutually exclusive.
2. Calculate probabilities for
events that have this
property.
Probability
Mutually Exclusive
www.mathsrevision.com
S4 General
Cannot
happen at
the same
time
When two events are MUTUALLY EXCLUSIVE
we can simply add their individual probabilities
P(A or B) = P(A) + P(B)
“or”
means add
29-Mar-16
Probability
Mutually Exclusive
S4 General
www.mathsrevision.com
Example :
Q.
Q.
Q.
There are 12 balls in a box.
4
12
3
What is the probability of picking a white ball. P(W) =
12
What is the probability of picking a green ball. P(G) =
What is the probability of picking a green OR white ball.
P(G or W) = P(G) + P(W)
29-Mar-16
4
3
7

+
=
12 12 12
Probability
Mutually Exclusive
www.mathsrevision.com
S4 General
Now try Ex 4A & 4B
Ch10 (page 127)
29-Mar-16
Starter Questions
www.mathsrevision.com
S4 General
1.
2.
3.
4.
Find the value of x if
5x =90
Calculate x o :
270
x
o
o
What is the probability of picking a red or black
card from a pack of cards.
Find the average of the numbers 9, 21, 40?
29-Mar-16
Probability
Independent Events
www.mathsrevision.com
S4 General
Learning Intention
1. To understand that if two
events are independent we
can simply multiply
probabilities.
29-Mar-16
Success Criteria
1. Understand when two events
are independent.
2. Calculate probabilities for
events that are independent.
Spinner 1
Probability
Independent Events
S4 General
www.mathsrevision.com
Spinner 2
Calculate
P(R) 1
Write down all the
possible outcomes
forare
spinner
1 =3
when the two spinners
spun together.
Spinner 2
Y
G
R
Y (Y,Y)
(Y,G)
(Y,R)
Spinner 1 G (G,Y)
(G,G)
(G,R)
R (R,Y)
(R,G)
(R,R)
Notice from = 1
table P (R,R) 9
29-Mar-16
Calculate (R) 1
=
for spinner 2 3
P(R,R) = P(R) x P(R)
Probability
Independent Events
S4 General
www.mathsrevision.com
Key Points
For two independent events A and B we can
multiply the individual probabilities
P(A and B) = P(A) x P(B)
Independent Events
1. List all combinations.
2. Calculate individual probabilities.
3. Multiply them together.
29-Mar-16
“and” means
multiply
Probability
Independent Events
www.mathsrevision.com
S4 General
Now try Exercise 5
Ch10 (page 129)
29-Mar-16