A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 o

Q: Wait, why is the total sample space $2^4$? Shouldn't it be more?

Think of it as four independent decisions. For each of the four positions in the matrix, you have exactly two choices: $0$ or $1$. Since these choices are independent, we multiply them: $2 \times 2 \times 2 \times 2 = 16$. It is the fundamental counting principle in action!

Q: Why can't the determinant be greater than $1$?

Excellent observation! Since $a, d \in \{0, 1\}$, the maximum value of the product $ad$ is $1 \times 1 = 1$. Since $b, c \in \{0, 1\}$, the minimum value of $bc$ is $0$. Therefore, the expression $ad - bc$ can only result in $1 - 0 = 1$, $1 - 1 = 0$, or $0 - 0 = 0$. It can never exceed $1$.

Q: Why do we multiply the number of ways for $ad=1$ and $bc=0$?

Because these are independent events! The choice of elements for the principal diagonal ($a$ and $d$) does not restrict the choices for the secondary diagonal ($b$ and $c$). When events are independent, we multiply their favorable outcomes to find the total number of ways the combined event can occur.

Question

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the value of determinant chosen is positive is .........

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with finding the probability that the determinant of a $2 	imes 2$ matrix, filled with entries from the set $\{0, 1\}$, is strictly greater than zero.

The matrix is defined as:
$$
\begin{pmatrix}
a & b \
c & d
\end{pmatrix}
$$

Since each of the $4$ positions has $2$ possible choices ($0$ or $1$), the total number of possible matrices is determined by the fundamental principle of counting:
$$
2 	imes 2 	imes 2 	imes 2 = 16
$$
This value, $16$, represents our total sample space.

### The Master Equation

The determinant of the matrix is given by the expression:
$$
\Delta = ad - bc
$$

We are searching for cases where $\Delta > 0$. Given that $a, b, c, d \in \{0, 1\}$, the product $ad$ can only be $0$ or $1$, and the product $bc$ can only be $0$ or $1$.

For the condition $\Delta > 0$ to hold, the only possible integer value for the determinant is $1$. This implies:
$$
ad - bc = 1
$$

### Evaluating Constraints

For the equation $ad - bc = 1$ to be satisfied, we must have $ad = 1$ and $bc = 0$.

First, consider the condition $ad = 1$. Since $a$ and $d$ are binary, the only way their product is $1$ is if:
$$
a = 1 \quad 	ext{and} \quad d = 1
$$
There is exactly **$1$ way** to satisfy this condition.

Next, consider the condition $bc = 0$. This is the complement of the case where $bc = 1$. Since there are $2 	imes 2 = 4$ total combinations for the pair $(b, c)$, and only one combination ($b=1, c=1$) results in $bc=1$, the number of ways to get $bc=0$ is:
$$
4 - 1 = 3
$$

### Final Calculation

By the rule of product, the total number of favorable outcomes is the product of the ways to satisfy the principal diagonal and the secondary diagonal:
$$
1 	imes 3 = 3
$$

The probability $P$ is the ratio of favorable outcomes to the total sample space:
$$
P = \frac{3}{16}
$$

The final probability is **$\frac{3}{16}$**.

Animated Solution for Mathematics - Probability: A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the value of determinant chosen is positive is .........

Visualized Solution

Defining the $2 \times 2$ Determinant

Calculating the Sample Space

Expanding the Determinant

Condition for a Positive Value

Solving for $a d = 1$

Solving for $b c = 0$

Counting Favorable Determinants

Calculating the Final Probability

The Sigma Insight: Classical Definition of Probability

Analyzing the Setup

The Master Equation

Evaluating Constraints

Final Calculation

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