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Example 33: A die is rolled, find the probability that an even number is obtained. Example 34: Two coins are tossed, find the probability that two heads are obtained. Note: Each coin has two possible outcomes H (heads) and T (Tils) Example 35: Two dice are rolled, find the probability that the sum is a) equal to 1 b) equal to 4 c) less than 13

Answer 1

Example 33:
let S be the sample space of rolling a dice

S={ 1 , 2 , 3 , 4 , 5 , 6 }
n(S)= 6,
Let A be the event that even number is obtained,
A={ 2 , 4 , 6 }
n(A)=3
By definition of the probability,
P(A)= n(A)/n(S)
= 3/6
=1/2
= 0.5

The probability that the even number is obtained is 0.5

Example 34:
Let S be the sample space of tossing two coins
S={ (HH), (HT) , (TH) , (TT) }
n(S)= 4
Let A be the event that two heads are obtained,
A={ (HH)}
n(A)=1
By definition of the probability,
P(A)= n(A)/n(S)
= 1/4
=0.25
The probability that the two heads are obtained is 0.25
Example: 35
let S be the sample space of rolling two dice

S={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(,3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(6,1)(6,2),(6,3),(6,4)}(6,5),(6,6)}
n(S)= 36,
a) Sum equal to 1:
Let A be the event that sum of the outcome is equal to 1,
A={ 0 }
n(A)=0
By definition of the probability,
P(A)= n(A)/n(S)
= 0/36
=0
The probability that the sum is equal to 1 is 0
b )Sum equal to 4 :
Let A be the event that sum of the outcome is equal to 4,
A={(1,3),(2,2),(3,1)}
n(A)=3
By definition of the probability,
P(A)= n(A)/n(S)
= 3/36
=1/12
= 0.8333

The probability that the sum is equal to 4 is 0.8333
C) Sum less than 13 :

Let A be the event that sum of the outcome is less than 13
A={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(,3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(6,1)(6,2),(6,3),(6,4)}(6,5),(6,6)}

n(A)=36
By definition of the probability,
P(A)= n(A)/n(S)
= 36/36
=1
The probability that the sum is less than 13 is 1