Transcript Reviews of probability

Reviews of probability
• Question 1:
• Suppose we have a “ABCDE” litters
how many words we can compose from
them with 4 litters without repetition.
5 
5!
5  4  3  2 1
P : 5P4  P(5,4)  P  

 120 words
1
 4  (5  4)!
r
n
Reviews of probability
• Question 2
• In a class there are 25 boys and 12 girls, the teacher
wants to select 1 boy and 1 girl to represent the
class for a function. In how many ways can the
teacher make this selection?
Solution:
• Here the teacher is to perform two operations as:
• Selecting a boy from among the 25 boys and
• Selecting a girl from among the 12 girls.
• Then the teacher makes this selection by
25 12  300
Reviews of probability
• Question 3
In how many different ways can a truefalse test consisting of 9 questions be
answered?
Reviews of probability
Answer:
• Each question has 2 choices True and
False and they are 9 questions so the
number of ways to answer the test is
2  512 wayes
9
Reviews of probability
• Question 4
• In how many ways a committee
consisting of 2 men and 1 woman, can
be chosen from 5 men and 3 women?
Solution:
Out of 5 men there are 2 men can be
chosen in 5C2 ways as:
Reviews of probability
Solution:
5C 2 
5!
5  4  3!

 10
2!(5  2)!
2!3!
Out of 3 women there are 1 women can be chosen as 3C1
ways as:
3!
3  2!
3C1 

3
1!(3  1)! 1!2!
Then the committee can be chosen in
5C2  3C1  10  3  30
Reviews of probability
• Question 5
• A die is loaded in such a way that an
even number is twice as likely to occur
as an odd number. If (E) is event that
the number equal or less than 3 occurs
in a single toss of the die.
Reviews of probability
• Answer
• 1. Find P(E)
• The sample space is S  {1,2,3,4,5,6}
S  {w,2w, w,2w, w,2w}
Now we assign a probability of w to each odd number
and a probability of 2w to each even number Then
E  {1 , 2 , 3 }  3
Reviews of probability
• E= {1, 2,3} <= 3
• P (E) = {w, 2w, w}
1 2 1 4
P( E )    
9 9 9 9
Let A be the event that an even number turns up and
let B be the event that a number divisive by 3 occurs. Find
P( A  B)
P( A  B)
Reviews of probability
• Solution: Event A is {2, 4, 6}, event B is
{3, 6}.
P( A  B)  {2,3,4,6}, P(A  B)  {6}
2 1 2 2 7
P( A  B)     
9 9 9 9 9
2
P( A  B) 
9
Reviews of probability
• Example 6:
• Two dice are tossed; find the
probability of getting an even number
on the first die or a total number of 8.
• Solution:
S  (6 2 )  36
A: Getting an even number on the first die.
B: The sum of the options obtained on the two dice 8.
Reviews of probability
1. A : {(2,4,6)  (1,2,3,4,5,6)}  n( A)  3  6  18
2. B : {( 2,6), (6,2), (3,5), (5,3), (4,4)}  n( B)  5
3. ( A  B) : {( 2,6), (6,2), (4,4)}  n( A  B)  3
4.
 P( A  B)  P( A)  P( B)  P( A  B) 
18 5
3 20



36 36 36 36
Reviews of probability
• Example 7:
Employed Unemployed Total
Male
460
40
500
Female
140
260
400
Total
600
300
900
If we select one person from above table and the event is
M: a man is chosen.
E: the once chosen is employed
Reviews of probability
• Solution: We can solve the above
problem by two methods as:
• Directly from table as:
460 23
P( M | E ) 

600 30
Reviews of probability
• Solution:2
number( E  M )
P( M | E ) 

number ( E )
P( E  M ) 
P( E ) 
number( E  M )
P( E  M )
nS

n( E )
P( E )
n( S )
460 23

900 45
600 2

900 3
23
23 3 69 23
P( M | E )  45 
 

2
45 2 90 30
3
Reviews of probability
• Example 8:
• In a fuse box containing 30 fuses of
which 6 are defective. If 2 fuses are
selected at random and removed from
the box without replacing the first, what
is the probability that both fuses are
defective?
Reviews of probability
• Solution:
• Let A be the event that the first fuse is
defective.
• Let B as the event that the A occurs
and then the B occurs after A has
occurred.
Reviews of probability
• Solution:
6 1
P ( A) 

30 5
5
P( B | A) 
29
1 5
1
P( A  B)  P( A).P( B | A)  

5 29 29