A round archery target of diameter 1 m is marked with four scoring regions from the centre outwards as red, blue, yellow and white. The radius of the red band is 0.20 m. The width of all the remaining bands is equal. If archers throw arrows towards the target, what is the probability that the arrows fall in the red region of the archery target ?

The correct option is 0.16.

To find the probability that an arrow falls in the red region of the archery target, we can use the concept of geometric probability, which is the ratio of the area of the desired region (the red region) to the total area of the target.

First, let's identify the dimensions of the archery target:
The diameter of the round archery target is given as 1 m.
Therefore, the total radius ($r_{\text{total}}$) of the target is:

$r_{\text{total}}=\frac{1\text{m}}{2}=0.5\text{m}$

Next, we calculate the total area of the archery target ($A_{\text{total}}$) using the formula for the area of a circle, $A=\pi r^2$:

$A_{\text{total}}=\pi {\left(0.5\right)}^2=0.25\pi {\text{m}}^2$

The red region is the innermost circular band. The radius of this red band is given as $r_{\text{red}}=0.20\text{m}$.
We calculate the area of the red region ($A_{\text{red}}$):

$A_{\text{red}}=\pi {\left(0.20\right)}^2=0.04\pi {\text{m}}^2$

Assuming that an arrow hitting the target is equally likely to land anywhere on its surface, the probability ($P$) that an arrow falls in the red region is the ratio of the area of the red region to the total area of the target:

$P=\frac{A_{\text{red}}}{A_{\text{total}}}$

Substituting the calculated areas into the formula:

$P=\frac{0.04\pi }{0.25\pi }$

The constant $\pi$ cancels out from both the numerator and the denominator:

$P=\frac{0.04}{0.25}=\frac{4}{25}=0.16$

Thus, the probability that the arrows fall in the red region of the archery target is 0.16.