Discrete Random Variables (DRV)

In a game of chance, a player has to spin a fair spinner. The spinner has \(7\) sections and an arrow which has an equal chance of coming to rest over any of the \(7\) sections. The spinner has \(1\) section labelled \(\mathbf{R}\), \(2\) sections labelled \(\mathbf{B}\) and \(4\) sections labelled \(\mathbf{Y}\) (see diagram).

Spinner diagram

The player then has to throw one of three fair six-sided dice, coloured red, blue or yellow. If the spinner comes to rest over \(\mathbf{R}\) the red die is thrown, if the spinner comes to rest over \(\mathbf{B}\) the blue die is thrown and if the spinner comes to rest over \(\mathbf{Y}\) the yellow die is thrown. The yellow die has one face with \(*\) on it and the blue die has two faces with \(*\) on it and the red die has three faces with \(*\) on it. The player wins the game if the die thrown comes to rest with a face showing \(*\) uppermost.

1. Find the probability that a player wins a game.[2]
2. Given that a player wins a game, find the probability that the spinner came to rest over \(\mathbf{B}\).[1]
3. Find the probability that a player wins \(3\) consecutive games, each time throwing a die of a different colour.[2]