Question

(a) From a stack of 3 dice, one is taken and rolled twice.
If, unknown to the gambler, two of the dice are weighted and each have a 1/10 chance of rolling a 6, what is the probability that the gambler rolls two 6's?
(b) If the gambler has rolled two sixes, what is the probability that he has rolled the weighted die?

Answer 1

(a) The probability of choosing the fair die is 1/3. If the fair die is chosen the probability of rolling two sixes is 1/6 * 1/6. The overall probability of choosing the fair die is
\[\frac{1}{3}\times\frac{1}{6}\times\frac{1}{6}=\frac{1}{108}\]
The overall probability of choosing a weighted die and rolling two sixes is
\[\frac{1}{3}\times\frac{1}{10}\times\frac{1}{10}=\frac{1}{300}\]
The probability of the gambler's rolling two 6's is
\[P(two\ 6's)=\frac{1}{108}+\frac{1}{300}+\frac{1}{300}=you\ can\ calculate\]

Answer 2

can i get help with b) @kropot72

Answer 3

(b) Given that two 6's have been rolled, the probability that a weighted die was rolled is given by\[P(a\ weighted\ die\ was\ rolled)=2\times(\frac{1}{300}\times\frac{2700}{43})=you\ can\ calculate\]