Variance of sum of independent values

3 A fair 8 -sided dice has faces labelled 10, 20, 30, ..., 80 .

1. State the distribution of the score when the dice is rolled once.
2. Write down the probability that, when the dice is rolled once, the score is at least 40 .
3. The dice is rolled three times.
   1. Find the variance of the total score obtained.
   2. Find the probability that on one of the rolls the score is less than 30 , on another it is between 30 and 50 inclusive and on the other it is greater than 50 .

5 A fair spinner has five faces, labelled 0, 1, 2, 3, 4.

1. State the distribution of the score when the spinner is spun once.
2. Determine the probability that, when the spinner is spun twice, one of the scores is less than 2 and the other is at least 2.
3. Find the variance of the total score when the spinner is spun 5 times.

7 The discrete random variable \(X\) has a uniform distribution over the set of all integers between 100 and \(n\) inclusive, where \(n\) is a positive integer with \(n > 100\).

1. Given that \(n\) is even, determine \(\mathrm { P } \left( \mathrm { X } < \frac { 100 + \mathrm { n } } { 2 } \right)\).
2. Determine the variance of the sum of 50 independent values of \(X\), giving your answer in the form \(\mathrm { a } \left( \mathrm { n } ^ { 2 } + \mathrm { bn } + \mathrm { c } \right)\), where \(a , b\) and \(c\) are constants.

A website awards a random number of loyalty points each time a shopper buys from it. The shopper gets a whole number of points between \(0\) and \(10\) (inclusive). Each possibility is equally likely, each time the shopper buys from the website. Awards of points are independent of each other.

1. Let \(X\) be the number of points gained after shopping once. Find
2. Let \(Y\) be the number of points gained after shopping twice. Find
3. Find the probability of the most likely number of points gained after shopping twice. Justify your answer. [4]