Transcript Probability - choosgs4math

n( A)
a set of desired outcomes

P(A) 
n( S ) a set of all possible outcomes
0  P( A)  1 and P( A ' )  1  P( A)
Ex 1
11 cards containing the letters of the word PROBABILITY is put
in a box. A card is taken out at random. Find the probability that
the card chosen is
(a) letter B
(b) a vowel
(c) a consonant
n( S )  11
(a)
n( B )  2
2
P( B) 
11
(b)
n(V )  4
4
P(V ) 
11
(c)
n(C )  7
7
P(C ) 
11
Ex 2
There are x red balls and 8 yellow balls in bag. A ball is taken
at random from the bag. The probability of getting a red ball is
3.
7
(a) Find the value of x.
Total number of balls = x + 8
x
3

x8 7
7 x  3x  24
4 x  24
x6
(b) If y red balls are then added to the
box, the probability of getting a yellow
ball becomes ½. Find the value of y.
Total number of balls = y + 14
8
1

y  14 2
y  14  16
y2
Ex 3
A box contains x orange sweets and 7 strawberry sweets.
If a sweet is taken at random from the box, the probability
of getting an orange sweet is 5 . Find the value of x.
12
5
P(orange sweet ) 
12
x
5

x  7 12
12 x  5x  35
7 x  35
x5
MUTUALLY EXCLUSIVE EVENTS

A
B
The probability of event A or event B
occurring / not mutually exclusive
P(A or B)
P( A  B)  P( A)  P( B)  P( A  B)
P ( A  B)  0

A
B
The probability of event A and event
B mutually exclusive.
P(A or B)
P( A  B)  P( A)  P( B)
because P( A  B)  0
METHOD
1 Find the possible outcomes of each event.
2 Find the possible outcomes of the sample space.
3 Find the probability of each event.
4
Determine if the events are mutually
exclusive. Keywords ‘or’ or ‘at least’.
5 Use the Addition Rule of Probability.
INDEPENDENT EVENTS
Two events are independent if the fact A occurs
does not affect the probability of B occurring.
P(A and B)
P( A  B)  P( A)  P( B)
METHOD
1 Find the probability of each event.
2 Determine if the events are dependent.
3 Use the Multiplication Rule of Probability to
calculate of both events.
Ex 4
Two dice, one black and the other is white, are tossed
together. Find the probability that
(a) an even number appears
on the black dice.
(b) the sum of the numbers
on the two dice is 7.
(c) an even number appears
on the black dice and the
sum of the numbers on
the two dice is 7.
(d) an even number appears
on the black dice or the
sum of the numbers on
the two dice is 7.
WD
● ● ● ● ● ●
6
5
4
● ● ● ● ● ●
3
2
1
● ● ● ● ● ●
0
1 2 3 4 5 6
● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ●
n(S )  36
BD
(a) A= events of even number appears on the black dice.
A  {( 2,1), (2,2),..., (2,6), (4,1), (4,2),..., (4,6), (6,1), (6,2),..., (6,6)}
18 1
P( A) 

36 2
(b) B = events with the sum of the numbers on the two dice is 7.
B  {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
6 1
P( B) 

36 6
(c) an even number appears on the black dice and the sum of the numbers on
the two dice is 7.
A  B  {( 2,5), (4,3), (6,1)}
3
1
P( A  B) 

36 12
(d) an even number appears on the black dice or the sum of the numbers on
the two dice is 7.
P ( A  B)  0
P( A  B)  P( A)  P( B)  P( A  B)
P( A  B)  1  1  1
2 6 12
7

12
Ex 5
Sarah is asked to write a number from the set
{ 1, 2, 3, 3,5 ,6}. Find the probability that she will write
(a) the number 3,
(b) the number 5,
(c) the number 3 or number 5
(a)
2 1
P(number 3)  
6 3
(b)
1
P(number 5) 
6
(c)
1 1
P(number 3 or number 5)  
3 6
1

2
Ex 6
The table shows the number of books on a book shelf.
Two books are taken from the shelf at random. Find the
probability that both books are of the same category.
Books
Price index in 2000
based on 1998
History
5
Geography
6
English
4
P(same category)  P( H , H )  P(G, G)  P( E, E )
5 4 6 5 4 3
      
15  14  15  14  15  14 
31

105
Ex 7
The probability that Hamid qualifies for the final of a track
event is 2 while the probability that Mohan qualifies is 1 .
5
3
Find the probability that
(a) both of them qualify for the final.
2 1
P( H  M )  
5 3
2

15
(b) only one of them qualifies for the final.
2 2 3 1
P( H  M ' )  P( H 'M )    
5 3 5 3
7

15
2
P( H ) 
5
HAMID
3
P( H ' ) 
5
11
M)) 
PP((M
33
MOHAN
2
H ') 
PP((M
3
Tree Diagram
a method of listing outcomes of an
experiment consisting of a series of
activities
Tree diagram for the experiment of
tossing two coins
H
H
start
T
H
T
T
Ex 8
Show the sample space for tossing one penny and rolling one die.
(H = heads, T = tails)
Solution:
By following the different paths in the tree diagram,
we can arrive at the sample space.
Sample space:
{ H1, H2, H3, H4, H5, H6,T1, T2, T3, T4, T5, T6 }
The probability of each of these outcomes
= (1/2)(1/6)
= 1/12
Ex 9
A family has three children. How many outcomes are in the
sample space that indicates the sex of the children? Assume that
the probability of male (M) and the probability of female (F) are
each 1/2.
Solution:
Sample space:
{ MMM
MMF
MFM
MFF
FMM
FMF
FFM
FFF }
There are 8
outcomes in the
sample space.
The probability of
each outcome
=(1/2)(1/2)(1/2)
= 1/8.
Short Quiz (MCQ):
Question 1:
You are at a carnival. One of the carnival games asks you to
pick a door and then pick a curtain behind the door.
There are 3 doors and 4 curtains behind each door. How
many choices are possible for the player?
(a) 3
(b) 12
(c) 24
(d) 36
Solution:
No. of possible
choices
= (3)(4)
= 12
Question 2:
The 4 aces are removed from a deck of cards. A coin is
tossed and one of the aces is chosen. What is the
probability of getting heads on the coin and the ace of
hearts?
Draw a tree diagram to illustrate the sample space.
(a) 1/2
Solution:
P (getting heads on coin and the
(b) 1/4
ace of hearts)
(c) 1/8
= (1/2)(1/4)
(d) 1/16
=1/8
Question 3:
There are 3 trails leading to Camp A from your starting
position. There are 3 trails from Camp A to Camp B. How
many different routes are there from the starting position
to Camp B? Draw a tree diagram to illustrate your answer
(a) 3
(b) 6
(c) 9
(d) 12
Solution:
No. of different routes
= (3)(3)
=9
Question 4:
A spinner has 4 equally likely regions numbered 1 to 4. The
arrow is spun twice. What is the probability that the spinner
will land on a 1 on the first spin and on a red region on the
second spin? Draw a tree diagram to represent your answer.
Solution:
(a) 1/2
(b) 1/4
(c )1/6
(d) 1/8
P(1 and red)
= (1/4)(1/2)
= 1/8
Question 5:
There are two identical bottles. One bottle contains 2 green
balls and 1 red ball. The other contains 2 red balls. A bottle
is selected at random and a single ball is drawn. What is the
probability that the ball is red?
(a) 1/2
(b) 1/3
(c) 1/6
(d) 2/3
Solution:
P(first bottle, red)
= (1/2)(1/3)
= 1/6
P(second bottle, red)
= (1/2)(1)
= 1/2
P(red)
= 1/6 + 1/2
= 2/3