1 When a four-sided spinner is spun, the number on which it lands is denoted by \(X\), where \(X\) is a random variable taking values \(2,4,6\) and 8 . The spinner is biased so that \(\mathrm { P } ( X = x ) = k x\), where \(k\) is a constant.

1. Show that \(\mathrm { P } ( X = 6 ) = \frac { 3 } { 10 }\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Kathryn is allowed three attempts at a high jump. If she succeeds on any attempt, she does not jump again. The probability that she succeeds on her first attempt is \(\frac { 3 } { 4 }\). If she fails on her first attempt, the probability that she succeeds on her second attempt is \(\frac { 3 } { 8 }\). If she fails on her first two attempts, the probability that she succeeds on her third attempt is \(\frac { 3 } { 16 }\). Find the probability that she succeeds.
4. Khaled is allowed two attempts to pass an examination. If he succeeds on his first attempt, he does not make a second attempt. The probability that he passes at the first attempt is 0.4 and the probability that he passes on either the first or second attempt is 0.58 . Find the probability that he passes on the second attempt, given that he failed on the first attempt.