Packets, nested counts and a modal-parameter range

A manufacturer produces a large number of coloured glowing sticks, which are packed in packets of 8. Each packet consists of randomly chosen coloured glowing sticks. On average, 12% of the glowing sticks are green.

(i) Explain the significance of ‘large number’ in the first sentence.

(ii) Find the probability that a randomly chosen packet of glowing sticks contains at least 3 green glowing sticks.

(iii) Ian buys 40 randomly chosen packets of glowing sticks for a concert. Find the probability that fewer than 6 of these packets contain at least 3 green glowing sticks.

On average, the proportion of glowing sticks that are blue is $p$. It is known that the modal number of blue glowing sticks in a packet is 3.

(iv) Use this information to find exactly the range of values that $p$ can take.