# Probability of what exactly was calculated by Chevalier de Méré?

Chevalier de Méré knew that probability of rolling six with a six-sided die is $\frac 16$. He reasoned that his chance to roll six with 4 tries is $\frac 46$, which is not the case. It is $1 - (\frac 56)^4$.

Which event's probability was calculated then? That is, which event has probability of $\frac 46$ in an experiment of throwing a six-sided die four times?

• He did not think that the probability of rolling at least one $6$ was $4/6$. He knew it was a slightly better than $50$-$50$ proposition, and asked about a double $6$ in $24$ rolls of two dice. He wondered whether the chances were the same. (The mean number of $6$ in $4$ rolls is $4/6$; the mean number of double $6$ in $24$ is $24/36$, which is the same as $4/6$. There was some confusion at the time about the relationship between probability and expectation.) – André Nicolas Feb 15 '16 at 7:04

There are several events that have a probability of $\frac46$ in that experiment. For example:

• Rolling a number smaller or equal to $4$ on the first throw
• Rolling a number larger or equal to $3$ on the second throw
• Rolling a number between $2$ and $5$ (inclusive) on the third throw

And on and on and on.

• He is asking an event which is relevant to the previous experiment.. – Win Vineeth Feb 15 '16 at 6:53
• @WinVineeth No, he isn't. You are assuming he is. And sorry, but in an experiment where you roll a die $4$ times, the event "I rolled a number smaller than $4$ on the first throw" is an event. – 5xum Feb 15 '16 at 6:54
• I know that you have given several events... If he was not asking for a relevant event, why will he quote the first sentence then? – Win Vineeth Feb 15 '16 at 6:56
• @5xum great, thanks! Can you come up with an example of an event that has probability of 4/6 and includes several throws (all four throws in the best case)? – Baldee Feb 15 '16 at 7:26
• @Baldee: An event in which the 1st throw is a number between $1-4$, the 2nd throw is a number between $1-6$, the 3rd throw is a number between $1-6$, and the 4th throw is a number between $1-6$ (which is exactly the description given in the first bullet above). – barak manos Feb 15 '16 at 7:47