Italian gambers used to bet on the total number of dots rolled with three dice.

Question

Italian gambers used to bet on the total number of dots rolled with three dice. they figured that tere are altogether six ways to obtain the sum 9 in three dice: 126, 135, 144, 234, 225, 333. Similarly, they found six ways to obtain sum ten: 145, 136, 226, 235, 244, 334. they argued that 9 and 10 should have the same chance, however, expereince showed 10 came up more often than 9.

A) whatr crucial aspect of this problem did the gamblers not realize?

b) list all of the outcomes on the event that the sum is 9?

C) list all the outcomes on the event that the sum is 10?

d) find the probabilities associated to the event that the sum is 9 and to the event that the sum is 10.

Explanation / Answer

A) what they did not realise or calculate is that the number of ways of getting each of the above combinations is different. for eg. to get 126 to get a sum of 9 can be done in 6 ways (126,162,216,261,612,621). Now this thinking will completely alter the chances of getting a 9 or 10 which then might not be equal. We shall illustrate it in the following 2 answers.

B) All outcomes where the sum is 9

126= (126,162,216,261,612,621) = 6 ways

135 = (135,153,315,351,513,531),=6 ways

144 = (144, 414 441) = 3 ways

234 = (234,243,324,342,423,432) = 6 ways

225 = (225,252,522) = 3 ways and

333. only 1 ways of getting this.

Therefore the total no of ways of getting 9 is = 6+6+3+6+3+1 = 25

C) All outcomes when the sum is 10

145 = (145,154,415,451,514,541) = 6 ways

136 = (136,163,316,361,613,631)= 6 ways

226 = (226,262,622) 3 ways

235 = (235,253,325,352,523,532) = 6 ways

244 = (244,424,442) = 3 ways

344 = (344,434,443) = 3 ways

Therefore total no of ways = 6+6+3+6+3+3 = 30 ways

So we see that there are 30 ways of getting a sum compared to 25 ways of getting a 9.

D) Probability of getting a sum of 9 and Probability getting a sum of 10.

P = Total no of favourable outcomes/ total no of outcomes

Total outcomes = 6*6*6 = 216 (there will be 216 ways in which the nos 1-6 will fall on these dice)

Therefore P(sum=9) = 25/216 = 0.116

and P(sum = 10) = 30/216 = 0.139

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