If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability ).
The order of determinant is 2 and each element can have two values 0 or 1. So the possible number of determinants is 2 × 2 × 2 × 2 = 16 .
The possible cases of determinant are,
| 1 0 0 0 | , | 0 1 0 0 | , | 0 0 1 0 | , | 0 0 0 1 | , | 1 0 0 1 | , | 1 1 0 0 | , | 1 0 1 0 | , | 0 1 0 1 | | 1 1 0 1 | , | 1 1 1 0 | , | 1 0 1 1 | , | 0 1 1 1 | , | 0 0 0 0 | , | 1 1 1 1 | , | 0 0 1 1 | , | 0 1 1 0 |
The determinant is positive in these 3 cases only,
| 1 0 0 1 | , | 1 1 0 1 | , | 1 0 1 1 |
The probability of positive value of determinant is,
P ( Positive value of determinant ) = 3 16
Thus, the required probability is 3 16 .
The order of determinant is 2 and each element can have two values 0 or 1. So the possible number of determinants is
The possible cases of determinant are,
The determinant is positive in these 3 cases only,
The probability of positive value of determinant is,
Thus, the required probability is
