Bayes Probability/ Conditional Probability

Question

A casino uses a regular die on Monday to Friday which is uniform on all faces and a weighted die on Saturdays and Sundays which is 50% likely to come out “6” and then 10% equally likely for the other rolls from 1 to 5. A player remember that they rolled a “6” but not the day played.

i. What is the chance that the player's "roll of a 6" came on Saturday?

ii. What is the chance that the “6” came from the regular die?

To use Bayes in this problem, how do you calculate the probability of a 6 regardless of the day it was rolled?

Answer 1

To use Bayes in this problem, how do you calculate the probability of a 6 regardless of the day it was rolled?

$$\mathbb{P}[X=6]=\frac{5}{7}\times \frac{1}{6}+\frac{2}{7}\times \frac{1}{2}$$

Answer 2

To use Bayes in this problem, how do you calculate the probability of a 6 regardless of the day it was rolled?

Represent the events with $W$ for a weekday, $U$ for a Sunday, $V$ for a Saturday, and $S$ for "success" (or "six"). Then just use the Law of Total Probability:$$\def\P{\mathop{\sf P}}\P(S)=\P(S\mid W)\P(W)+\P(S\mid U)\P(U)+\P(S\mid V)\P(V)$$