Multiple independent trials or dice

A question is this type if and only if it involves rolling dice or spinning spinners multiple times independently and asks about combined outcomes, totals, or specific patterns.

2 questions

OCR MEI Further Statistics Major 2022 June Q9
9 The random variable \(X\) has a discrete uniform distribution over the values \(\{ 0,1,2 , \ldots , 20 \}\).
  1. Find \(\mathrm { P } ( X \leqslant 7 )\).
  2. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    The spreadsheet shows a simulation of the distribution of \(X\). Each of the 25 rows of the spreadsheet below the heading row shows a simulation of 10 independent values of \(X\) together with the value of the mean of the 10 values, denoted by \(Y\).
    \includegraphics[max width=\textwidth, alt={}]{77eabbd6-a058-457f-9601-d66f3c2db005-07_38_45_880_279}ABCDEFGHIJKL
    1\(X _ { 1 }\)\(X _ { 2 }\)\(X _ { 3 }\)\(X _ { 4 }\)\(X _ { 5 }\)\(X _ { 6 }\)\(X _ { 7 }\)\(X _ { 8 }\)\(X _ { 9 }\)\(X _ { 10 }\)\(Y\)
    216211864911116.9
    313141224111601608.8
    441711641012218139.7
    5281214161221588.0
    6715160471130208.3
    71513101120201516610.8
    81413171221816189412.3
    9202123173018151310.3
    10212512260910157.3
    115111310917104201511.4
    12149976202211169.6
    1315191819766203812.1
    1451064119158171810.3
    150315151112039168.4
    16112115041111926.6
    171250838121913129.2
    1895113541811197.6
    19162202012172782012.4
    20181732818701169.0
    211510720405611149.2
    223910142186076.0
    23111011101911371009.2
    241214665201118101411.6
    25111514111011205.6
    26014711185102011910.5
    27
  3. Use the spreadsheet to estimate \(\mathrm { P } ( Y \leqslant 7 )\).
  4. Explain why the true value of \(\mathrm { P } ( Y \leqslant 7 )\) is less than \(\mathrm { P } ( X \leqslant 7 )\), relating your answer to \(\operatorname { Var } ( X )\) and \(\operatorname { Var } ( Y )\).
  5. The random variable \(W\) is the mean of 30 independent values of \(X\). Determine an estimate of \(\mathrm { P } ( W \leqslant 7 )\).
Edexcel S1 2003 June Q5
5. The random variable \(X\) represents the number on the uppermost face when a fair die is thrown.
  1. Write down the name of the probability distribution of \(X\).
  2. Calculate the mean and the variance of \(X\). Three fair dice are thrown and the numbers on the uppermost faces are recorded.
  3. Find the probability that all three numbers are 6 .
  4. Write down all the different ways of scoring a total of 16 when the three numbers are added together.
  5. Find the probability of scoring a total of 16 .