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 \(\mathbb{E}(S)\).
Suppose that three fair 6-sided dice are rolled.
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Posted in Dice Rolls, Expectations, Max and Mins, Tail Sum Fromula
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.
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Posted in Bayes Theorem, Binomial, Conditioning, Dice Rolls
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?
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Posted in Addition rule, Counting, Dice Rolls
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.
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Posted in Dice Rolls, Independence
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.
[ from Meester, ex 1.7.32]
Let \(X_1,…,X_5\) be five iid rolls of six sided die. Let \(Z = \mbox{max}\{X_1,…,X_5\}\). Compute \(\mathbf{E}(Z)\).
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Posted in Dice Rolls, Max and Mins
A fair die is rolled ten times. Find numerical values for the expectations of each of the following random variables
[From Pitman page 183]
Posted in Dice Rolls, Expectations
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]
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Posted in Dice Rolls, Max and Mins, Sequence of independent trials
A standard 6 sided die is rolled three times.
Based on [Pitman, p. 182 #3]
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Posted in Dice Rolls, Expectations
Tagged JCM_math230_HW56_F22, JCM_math230_HW5_S13, JCM_math230_HW5_S15, JCM_math340_HW5_F13
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 \(Y_2=\min(X_1,X_2)\).
Combination of [Pitman, p. 159 #4 and #5]
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Posted in Dice Rolls, Max and Mins
Tagged JCM_math230_HW4_S13, JCM_math230_HW56_F22, JCM_math230_HW5_S15, JCM_math340_HW4_F13