Answered: A bag contains three kinds of dice:…

Holt Mcdougal Larson Pre-algebra: Student Edition 2012

1st Edition

ISBN:9780547587776

Author:HOLT MCDOUGAL

Publisher:HOLT MCDOUGAL

Chapter11: Data Analysis And Probability

Section11.8: Probabilities Of Disjoint And Overlapping Events

Problem 2C

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A bag contains three kinds of dice: seven 4-sided dice, three 6-sided dice, and two 12-sided dice. A die is drawn from the bag and then rolled, producing a number. For example, the 12-sided die could be chosen and rolled, producing the number 10. Assume that each die is equally likely to be drawn from the bag.

a. What is the probability that the roll gave a six?

b. What is the probability that a 6-sided die was chosen, given that the roll gave a six.

Expert Solution

Step 1

Let X: the number obtained is 6

A: A four sided dice is drawn

B: A six sided dice is drawn

C: A twelve sided dice is drawn

We know,

$P (A) = \frac{7}{12} P (B) = \frac{3}{12} P (C) = \frac{2}{12}$

And

$P (X | A) = 0 P (X | B) = \frac{1}{6} P (X | C) = \frac{1}{12}$

Using Law of Total Probability,

$P (X) = P (X | A) \times P (A) + P (X | B) \times P (B) + P (X | C) \times P (C) = 0 + \frac{1}{6} \times \frac{3}{12} + \frac{1}{12} \times \frac{2}{12} = \frac{1}{18}$

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