Question

If $a_1$, $a_2$, $b_1$ and $b_2$ take values in the set $\{1, -1, 0\}$, then the probability that the equation $a_1a_2 = b_1b_2$ is satisfied, is $\frac{p}{q}$, (where $p$ and $q$ are co-prime positive integers), then $q + 2p$ is equal to ___.

Answer 1

49

Answer 2

There are 3 choices for each of the four variables, so the total number of outcomes is

$$3^4 = 81.$$

Consider the pair $(a_1, a_2)$. The product $a_1a_2$ can be:

• $1$ in 2 cases: $(1, 1)$ and $(-1, -1)$,
• $-1$ in 2 cases: $(1, -1)$ and $(-1, 1)$,
• $0$ in 5 cases: $(1,0), (0,1), (-1,0), (0,-1)$ and $(0,0)$.

Similarly, the pair $(b_1, b_2)$ has the same distribution.

For the equation $a_1a_2 = b_1b_2$ to hold, both products must be equal. The number of favorable outcomes is:

$$\text{Favorable} = 2^2 + 2^2 + 5^2 = 4 + 4 + 25 = 33.$$

Thus, the probability is $\frac{33}{81} = \frac{11}{27}$.

Here, $p = 11$ and $q = 27$. The required value is:

$$q + 2p = 27 + 2(11) = 27 + 22 = 49.$$