In the 1650\'s Chevalier de Mere and Blaise Pascal (with Pierre de Fermat acting
ID: 3202058 • Letter: I
Question
In the 1650's Chevalier de Mere and Blaise Pascal (with Pierre de Fermat acting as a consultant) discussed the following question. (They were developing the foundations of probability theory. Although Pascal and Fermat, in particular, were smart people, this and similar questions were tricky to figure out at this early time.) Which of the following is more likely? (Calculate the probabilities to answer the question.) In four throws of a single die to get at least one six. In twenty-four throws of two dice to get at least one double six. Since 4/6 = 24/36, one naively says that they are equally likely. Expose the flaw in reasoning.Explanation / Answer
A:
In a roll of a die, there are six possible outcomes: 1, 2, 3, 4, 5, 6. If the die is fair, the probability of getting a six is 1/6. Likewise, the probability of getting no six in one roll of a fair die is 5/6.
The probability of getting no six in four rolls is : 5/6 x 5/6 x 5/6 x 5/6 = 625/1296 = 0.482253
Thus in four rolls of a fair die, the probability of getting at least one six is: 1 - The probability of getting no six in four rolls = 1- 0.482253 = 0.517747
B:
In a roll of a pair of dice, there are a total of 36 possible outcomes (i.e. the six outcomes of the first die combined with the six outcomes of the second die). Out of these 36 outcomes, only one of them is a double six. So, the probability of getting a double six is 1/36 in rolling a pair of dice. Likewise, the probability of not getting a double six is 35/36.
The probability of getting no double six in 24 rolls of a pair of dice is: = (35/36)24 = 0.5086
The probability of getting atleast one double six in 24 rolls of a pair of dice = 1 - The probability of getting no double six in 24 rolls of a pair of dice = 1- 0.5086 = 0.4914
Finally
In four rolls of a fair die, the probability of getting at least one six = 0.517747 is greater than
The probability of getting atleast one double six in 24 rolls of a pair of dice = 0.4914
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