Transcript Probability2009-09

MATHEMATICS
Probability
MAIN TOPIC
 Suppose that a bag contains 3 ref balls and 2 white
balls, then a ball is drawn on random. Since the bag
contains 5 balls, then there are 5 possible balls to be
drawn. We write down: n(S) = 5, which means, the
number of all possibilities is 5. Next suppose that the
event when we draw the red ball is E.
 Since there are 3 red balls that are possible to be
drawn, then we write down n(E) = 3.
 The probability of the event E, written as P(E), is
defined by the formula:
n( F )
P( E ) 
n( S )
 So in our example :
 P(E) = 3
5
 r 0,6 which means, the probability that a red ball is
drawn is 0,6. By the same understanding, if F is the
event when the white ball is dawn, so that
n( F ) 2
P( F ) 

n( s ) 5
 or 0,4. Which means, the probability that a white ball is
drawn is 0,4.
 You may imagine that probability of an event is the
level of certainty of that event to be happened. The
probability = 0 indicates that an event is not possible to
be happened (mathematically), while the probability =
1 indicates that an event must be happen
(mathematically).
 The value of probability range between 0 – 1 and the
sum of probabilities of all event in one case (a case is
also called an experiment) is 1.
3 2
P
(
E
)

P
(
F
)

 1
 For instance in our previous example:
5 5
 since E and F are all event in the mentioned case
(experiment) that is the case when a ball is drawn on
random from the bag contains 3 red balls and 2 white
balls. And since P(E) + P(F) = 1, then P(F) = 1 – P(E).
 The event F is the complement of the event E, or the
event F happens when the E does not happen (notice
that the event E and F do not happen at the same
moment). So, the probability of a complement of an
event = 1 – the probability of that event.
 Complement of the event E can be written down as
so
E
E
that and we obtain: p( E ) = 1 – P(E) or P(E)
E = 1 – P (
) In our everyday life, we often use the concept of
probability, actuality, and also any other men that does
know at all about the theory of probability. For instance,
a wresling commentator say that the probability T – ice
wins is 30%.
30
3

 What he wants to say is: probability = 100 10
 Example 22
 A coin is tossed. Find the probability that head turns up.
 Answer: Explanation: Here n(S) = 2, since there are 2
possible out comes, that is head or tail. If E is event
when head trust up, then n(E) = 1, since a coin has only
one head so that P(E) = n( E )  1
n( S )
2
 Notice also that complement of the event E, that is , is
the event when tail turns up.
Example 1
If a dice is thrown, then find the probability that
an event number is scored.
Answer
3
1
p 
or
6
2 n(S )  6, n( E)  3; E  2,4,6
Example 24
There coins are tossed. Find the probability that I
head and 2 tails turn ip.
Answer:
A = head
G = tail
3
n(S) = 8, n(E) = 3 marked by* , so that P(E) =
8


Example 2
If two dice are thrown at the same moment, find the
probability that a total of 5 is scored.
Answer:
Dice I
Dice II
1
2
3
4
5
6
n(S) = 6x6= 36
n(E) = 4
P(E) =
1
2
3
4
5
6
(1,1)
(2,1)
(3,1)
(4,1)
(5,1)
(6,1)
(1,2)
(2,2)
(3,2)
(4,2)
(5,2)
(6,2)
(1,3)
(2,3)
(3,3)
(4,3)
(5,3)
(6,3)
(1,4)
(2,4)
(3,4)
(4,4)
(5,4)
(6,4)
(1,5)
(2,5)
(3,5)
(4,5)
(5,5)
(6,5)
(1,6)
(2,6)
(3,6)
(4,6)
(5,6)
(6,6)
4 1

36 9
Example 3
 If the probability that Ani does not pass an exam is
35% that find the probability that Ani passes the exam.
 Answer:
1
35
 0,65or100%  35%  65%
100
 since passes and does not pass are complementary.
Example 4
 Find the probability that a family of 3 children has 1 son
and two daughters.
 Answer:
 If A = son and B = daughter, then diagram is the same
as in example 24.
AGA
GAA
n(S) = 8
AGG*
GAG*
n(E) = 3
AAA
GGA*
p(E) = 3
8
AAG
GGG
Thank You