Die Wager II

Alice, Bob, and Carlos are on a game show where they each roll a fair standard $(4n)-$sided die in private, where $n \geq 1$. Assume each person only takes bets when they know they have larger than $50\%$ chance of winning the bet.

• Alice responds no to the question “Would you like to bet that your number is higher than or equal to Carlos’?"

• Carlos responds yes to the question “Would you like to bet that your number is higher than or equal to Alice's?”

Afterwards, the host asks Bob, who rolled the value $k > 2n$, "Would you like to bet that your number is higher than or equal to Carlos’?" If Carlos heard the conversation with Alice, what is the probability Bob wins the bet? Report the answer with $n = 5$ and $k = 14$.