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Coursework questions

1.
A firing squad is composed of three policemen, Abel, Baker and Charlie. Each policeman shoots, but they only have probabilities $0.6$ , $0.7$ and $0.8$ , respectively, of hitting the victim. Furthermore only one of the three bullets is live, and is allocated at random. Find
1. (a)
  
  the probability that the victim is hit by the live bullet,
2. (b)
  
  the probability that, if the victim was hit by the live bullet, it was Charlie who had the live round.
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2.

Two balls are randomly chosen, without replacement, from an urn containing 8 blue, 4 yellow and 2 green balls. Suppose that we win £2 for each yellow ball selected and lose £1 for each blue ball selected. Let $R$ be the random variable giving our profit. Find the induced sample space for $R$ and evaluate the probabilities on the induced sample space.

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3.

Show that the two definitions of expectation,

$\displaystyle{\rm E}[R]=\sum_{r=0}^{\infty}rp(r)\quad\mbox{and}\quad{\rm E}[R]% =\sum_{\omega\in\Omega}R(\omega){\rm P}(\{\omega\}),$

are equivalent.

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