A blind box series named MathBots includes $4$ bots and $4$ number sets as accessories for the b

Question

A blind box series named MathBots includes $4$ bots and $4$ number sets as accessories for the bots. Bots are named $\alpha $, $\beta $ and $\gamma $ with $\omega $ as the special character bot.

The number sets are $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{N}$. Each box of the series contains one of the bots and one of the number sets.

The data in the table below shows the probabilities of getting boxes with the specified contents.

	$\mathbb{R}$	$\mathbb{Z}$	$\mathbb{Q}$	$\mathbb{N}$
$\alpha$	$\frac{1}{8}$	$\frac{1}{8}$	$0$	$\frac{1}{20}$
$\beta$	$0$	$\frac{1}{8}$	$\frac{1}{8}$	$\frac{1}{20}$
$\gamma$	$\frac{1}{8}$	$0$	$\frac{1}{8}$	$\frac{1}{20}$
$\omega$	$0$	$0$	$0$	$\frac{1}{10}$

(For example, there is a $\frac{1}{10}$ probability that a randomly chosen box contains $\omega $ and $\mathbb{N}$.)

A collector randomly chooses $4$ boxes for purchase.

Show that the probability of the collector getting all the bots and number sets is $\frac{3}{320}$.[2]
Find the probability that the collector gets all the bots given he manages to get all the number sets.[3]
Determine if the event that he gets all the bots and the event that he gets all the number sets are independent.[2]

Accepted Answer

(i) $\dfrac3{320}$ (ii) $\dfrac1{10}$ (iii) The events are not independent

	\(\mathbb{R}\)	\(\mathbb{Z}\)	\(\mathbb{Q}\)	\(\mathbb{N}\)
\(\alpha\)	\(\frac{1}{8}\)	\(\frac{1}{8}\)	\(0\)	\(\frac{1}{20}\)
\(\beta\)	\(0\)	\(\frac{1}{8}\)	\(\frac{1}{8}\)	\(\frac{1}{20}\)
\(\gamma\)	\(\frac{1}{8}\)	\(0\)	\(\frac{1}{8}\)	\(\frac{1}{20}\)
\(\omega\)	\(0\)	\(0\)	\(0\)	\(\frac{1}{10}\)

2023 RVHS P2 Q8

2023 RVHS P2 Q8

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