The system has a corresponding augmented matrix
Let's get this into reduced row echelon form.
Add 3(row 1) to -2(row 2), and add 5(row 1) to -2(row 3):
Multiply through row 2 by -1:
Add -1(row 2) to row 3:
Multiply through row 3 by 1/2:
Add row 3 to row 2:
Add -3(row 2) and row 3 to row 1:
Multiply through row 1 by 1/2:
This tells us
The matrix has rank 3, so we can pick 2 free variables, say and
. Then the general solution to the system is
where .
The required probability is 1 - 0.330 = 0.6697 .
P ( x= 2 ) = 1 /6 [ p , probability of success ]
q = 1 - ( 1 / 6 )
= 5 / 6
n = 84
Mean = np
= 84 * ( 1 / 6 )
= 14
npq = 14 * ( 5 / 6 )
= 11.667
Applying normal approximation ,
To calculate P ( x < 12 ) = P { Z < [ ( 12 - 14 ) / ( ) ] }
= P [ Z < ( - 2 ) / ( 3.416 ) ]
P [ Z < - 0.585 ] = 0.279
Therefore , the probability 2 occurs more than 12 times = 1 - 0.279
= 0.721
Applying continuity correction ,
P ( x < = 12 ) = P ( x < 12.0 + 0.5 ) = P ( x < 12.5 )
= P [ Z < ( 12.5 - 14 ) / 3.416 ]
= P ( Z < - 0.439 )
= 0.330
Hence , the required probability is 1 - 0.330 = 0.6697 .
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