Transcript Lesson One Introduction to Probability Theory File

An Introduction to
Probability Theory
Simple Probability
Simple probability is defined as:
P(event) 
Number of ways the event can occur
Total possible number of outcomes of the trial
n( A)
P( A) 
n(U )
A probability of an event is certain to occur is 1
and the probability that an event cannot occur is 0.
0  P( E )  1
P (U )  1
If the probability of an event occurring is p, then the
probability of this event not occurring is (1 – p)
P ( E ')  1  P( E )
P( A)  P( A ')  1
If E  F , then P( E )  P( F )
E and E’ are Complementary Events
Probability (Possibility) Space
A number is selected at random from the set {2, 4, 6, 8} and another
number is selected from the set {1, 3, 5, 7}. The two numbers are
multiplied together. Draw a sample space diagram.
2
2
6
10
14
4
4
12
20
28
6
6
18
30
42
8
8
24
40
56
1
3
5
7
A bag contains 8 disks of which 4 are red, 3 are blue and 1 is
yellow. Calculate the probability that when one disk is drawn
from the bag it will be:
a) Red
Answer:
c) Blue
P(Red) 
1
2
Answer:
b) Yellow
Answer:
P(Blue) 
3
8
d) Not blue
P(Yellow) 
1
8
Answer: P(Not Blue) 
5
8
You can use Set notation when evaluating probabilities
Example:
One element is randomly selected from a universal set of 20
elements. Sets A and B are subsets of the universal set and
n(A) = 15, n(B) = 10 and n(A  B)  7. Find:
c) P(A’)
a) P(A)
Answer:
b)
P(A) 
3
4
Answer:
P (A  B)
Answer:
P(A  B) 
d)
7
20
1
P(A') 
4
P (A  B)
Answer:
P(A  B) 
9
10
A and B are subsets of the universal set and n(A) = 25,
n(B) = 20, [n(A  B) ']  20 and there are 50
elements in the universal set. When one element is
selected at random, calculate:
a) P(A)
Answer:
b)
c) P(B’)
3
Answer: P(B') 
5
1
P(A) 
2
P (A  B)
Answer: P(A  B) 
d)
3
5
P (A  B)
Answer: P(A  B) 
3
10
A number k is chosen from {-3, -2, -1, 0, 1, 2, 3, 4}. What is the
probability that the expression below can be written as a product of two
linear factors, each with integer coefficients?
x2  2 x  k
Answer:
3
8
A number c is chosen from {1, 2, 3, 4, 5, 6}. What is the probability that
the expression below intersects the x-axis?
y  x2  4 x  c
Answer:
2
3
A secretary has three letters to put into envelopes. Being in a rush, she
puts them in at random. Find the probability that:
a) Each letter is in the correct envelope
b) No letter is in its correct envelope.
Answers:
1
6
a)
b) 13
Two dice are rolled and the product of the scores is found. Find the
probability that the product is:
a) odd
b) prime
Answers:
a)
b) 16
1
4
Three people, A, B and C play a game which is purely determined by
chance. Find the probability that they finish in the order ABC.
Answer:
1
6
Two dice are rolled. Find the probability that:
a) the total score is 10
b) the dice show the same number
Answers:
a)
b) 16
1
12
There are 30 students in a class, of which 18 are girls and 12 are
boys. Four students are selected at random to form a committee.
Calculate the probability that the committee contains
(a)
two girls and two boys;
(b)
students all of the same gender.
Answer:
(a) 0.368
(b) 0.130
SPEC06/HL1/15