Sophie and Ella play a
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30 +dyh dx;-r u04zvzd auuza5 h38xz.qgyfl7able in order I, V, X, L, C as shown in the following diagram.
Ella shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Sophie's 4 card directly above. Sophie wins if no matches occur; otherwise Ella wins.
1. Show that the probability that Sophie wins the game is $\frac{11}{30}$ .
Sophie and Ella repeat their game so that they play a total of 90 times. Let the discrete random variable X represent the number of times Sophie wins.
2. Determine:
1. the mean of X ;
2. the variance of X .