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Minimum Dice Roll
Suppose that three fair 6-sided dice are rolled.
- Let \(M\) be the minimum of three numbers rolled. Find \(\mathbb{E}(M)\).
- Let \(S\) be the sum of the largest two rolls. Find \(\mathbb{E}(S)\).
Loaded Dice
You have a pair of fair dice and a pair of loaded dice. But you forgot which pair is which. You do remember that when you bought the loaded dice, the company that makes them claimed the dice would land on a sum of 7 approximately 1/3 of the time.
- You choose one of the pairs at random and roll it once. You get a sum of 7. What is the likelihood that you picked the loaded dice?
- You choose one of the pairs at random and roll the pair three times. You get exactly one sum of 7. What is the likelihood that you picked the loaded dice?
Dice rolling addition rule
You roll a fair 6-sided die 3 times. What is the likelihood of getting exactly one 4, exactly one 5, or exactly one 6?
Dice Rolling Events
Consider rolling a fair 6-sided die twice. Let \(A\) be the event that the first roll is less than or equal to 3. Let \(B\) be the event that the second roll is less than or equal to 3. Find an event \(C\) in the same outcome space as \(A\) and \(B\) with \(0<\mathbb{P}(C)<1\) and such that \(A\), \(B\) and \(C\) are mutually independent, or show that no such event exists.
Prime Dice
Suppose that we have a very special die which has exactly \(k\) faces where \(k\) is prime. The faces are numbered \(1,\dots,k\). We throw the die once and see which number comes up.
- What would be an appropriate outcome space and probability measure for this random experiment ?
- Suppose that the events \(A\) and \(B\) are independent. Show that \(\mathbf{P}(A)\) or \(\mathbf{P}(B)\) is always either 0 or 1. Or in other wards \(A\) or \(B\) is always either the full space or the empty set.
[ from Meester, ex 1.7.32]
Maximum of die rolls
Let \(X_1,…,X_5\) be five iid rolls of six sided die. Let \(Z = \mbox{max}\{X_1,…,X_5\}\). Compute \(\mathbf{E}(Z)\).
Expectations of die rolls
A fair die is rolled ten times. Find numerical values for the expectations of each of the following random variables
- the sum of the numbers in the ten rolls;
- the sum of the largest two numbers in the first three rolls;
- the maximum number in the first five rolls;
- the number of multiples of three in the first ten rolls;
- the number of faces which fail to appear in the ten rolls;
- the number of different faces that appear in the ten rolls.
[From Pitman page 183]
Two die
Two dice are rolled. Find the probabilities of the following events.
a) the maximum of the two numbers rolled is less than or equal to 2;
b) the maxinum of the two numbers rolled is less than or equal to 3;
c) the maximum of the two numbers rolled is exactly equal to 3;
d) Repeat b) and c) with 3 replaced by \(x=1,…,6\);
e) Denote \( \mathbf{P}(x)\) as the probability that the maximum number is exactly \(x\).
Compute \( \sum_{x=1}^6\mathbf{P}(x)\).
[Pitman Page 10, #7]
Expection and dice rolls
A standard 6 sided die is rolled three times.
- What is the expected value of the first roll ?
- What is the expected values of the sum of the three rolls ?
- What is the expected number of twos appearing in the three rolls ?
- What is the expected number of sixes appearing in the three rolls ?
- What is the expected number of odd numbers ?
Based on [Pitman, p. 182 #3]
Dice rolls: Explicit calculation of max/min
Let \(X_1\) and \(X_2\) be the number obtained on two rolls of a fair die. Let \(Y_1=\max(X_1,X_2)\) and \(Y_2=\min(X_1,X_2)\).
- Display the joint distribution tables for \( (X_1,X_2)\).
- Display the joint distribution tables for \( (Y_1,Y_2)\).
- Find the distribution of \(X_1X_2\).
Combination of [Pitman, p. 159 #4 and #5]
