Answer by joriki for A standard 6-sided fair die is rolled until the last 3...
This can be modelled using a Markov chain. We need $15$ initial states: One initial state in which we have nothing (either because we just started or because the last roll didn’t leave enough room for...
View ArticleAnswer by Eric Brier for A standard 6-sided fair die is rolled until the last...
The process of rolling the dice till getting an increasing triplet is well modelized by a Markov chain: the current state consists in the last three dice rolls, which gives $6^3=216$ possibilities.We...
View ArticleAnswer by dan_fulea for A standard 6-sided fair die is rolled until the last...
I must draw a diagram with all "relevant paths" toWIN - last four rolls are in order $1234$LOSS / GAME OVER - last four rolls are in order among $1235, 1236, 1345, 1346, 1356, 1456, 2345, 2346, 2356,...
View ArticleA standard 6-sided fair die is rolled until the last 3 rolls are ascending....
A standard $6$-sided fair die is rolled until the last $3$ rolls are ascending. What is probability that the last $3$ digits start with $1,2,3,4$ respectively?My attemptWe can investigate the $3-$roll...
View ArticleAnswer by Dan for A standard 6-sided fair die is rolled until the last 3...
A simple Monte Carlo method:import randomNUM_SIMULATIONS = 10**6def roll_die(): return random.randrange(1, 7)def roll_until_ascending(): dice = [] while len(dice) < 3 or not (dice[-3] < dice[-2]...
View ArticleAnswer by Dan for A standard 6-sided fair die is rolled until the last 3...
A simple Monte Carlo method:import randomNUM_SIMULATIONS = 10**6def roll_die(): return random.randrange(1, 7)def roll_until_ascending(): dice = [] while len(dice) < 3 or not (dice[-3] < dice[-2]...
View ArticleAnswer by joriki for A standard 6-sided fair die is rolled until the last 3...
This can be modelled using a Markov chain. We need $15$ transient states: One initial state in which we have nothing (either because we just started or because the last roll didn’t leave enough room...
View ArticleAnswer by Eric Brier for A standard 6-sided fair die is rolled until the last...
The process of rolling the dice till getting an increasing triplet is well modelized by a Markov chain: the current state consists in the last three dice rolls, which gives $6^3=216$ possibilities.We...
View ArticleAnswer by dan_fulea for A standard 6-sided fair die is rolled until the last...
The modelling probability space is the space of all infinite tuples with entries among $1,2,3,4,5,6$. There is a stopping time $\tau$, the first occurrence of the pattern$$a<b<c$$with three last...
View ArticleA standard 6-sided fair die is rolled until the last 3 rolls are strictly...
A standard $6$-sided fair die is rolled until the last $3$ rolls are strictly ascending. What is probability that the first such roll is a $1$, $2$, $3$, or $4$?My attemptWe can investigate the...
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