Question

Suppose you roll two dice, one black, and one red. You record the outcomes in order. For example, the outcome (35) denotes a 3 on the black die and a 5 on the red die. Let A be the event that the black die is even, B be the event that the red die is odd, and C is the event that the dice sum to 10.

(a) (4pts) List or describe the outcomes in the sample space S and the events in A, B, and C. Report the number of outcomes in each set.

(b) (1pt) Find P(A), P(B), and P(C).

(c) (2pts) Find P(A and B). Are A and B independent?

(d) (2pts) Find P(A and C). Are A and C independent? 

Answer 1

Solution.

а) Sample space: set of all possible combinations of rolling two dices in form (B, R), where B, R - numbers [1; 6]

A - {(2, 1), (2, 2), ... , (4, 1), (4, 2), (4, 3), ..., (6, 1), ...} ... means all other combinations of even numbers in the first place and any number in the second place. Total number of elements: 3*6=18

B - {(1, 1), (2, 1), ..., (1, 3), (2, 3), (3, 3), ..., (1, 5), ...} ... means all other combinations of odd numbers in the second place and any number in the first place. Total number of emenets: 6*3=18

C - {(4, 6), (6, 4), (5, 5)}. Total number of elements: 3

b) There are 3*6=18 elements in set A. Total number of outcomes is 6*6 = 36

$P(A) = \frac{18}{36}=\frac{1}{2}=0.5$

There are also 18 elements in set B, so:

$P(B) = \frac{18}{36}=0.5$

For the C:

$P(C) = \frac{3}{36} = \frac{1}{12} \approx 0.083$

c) Total number of (A and B) elements is 3*3=9

$P(A\land B) = \frac{9}{36} = \frac{1}{4}=0.25$

For A and B to be independent, the following condition must work:

$P(B|A) = P(B)$, and

$P(B|A) = \frac{P(A\land B)}{P(A)} = \frac{0.25}{0.5} = 0.5 = P(B)$

Therefore, those events are independent.

d) Total number of elements in (A and C) is 2, so:

$P(A \land C) = \frac{2}{36} = \frac{1}{18}$

For A and C to be independent, we need:

$P(A|C) = P(A)$

$P(A|C) = \frac{P(A \land C)}{P(C)} = \frac{1/18}{1/12} = \frac{12}{18}=\frac{2}{3}\neq P(A)$

Therefore, these events are not independent

Answer:

a) Described in solution

b) $P(A) = P(B) = 0.5, P(C) =\frac{1}{12}$

c) $P(A\land B) = 0.25$, events A and B are independent

d) $P(A \land C)=\frac{1}{18}$, events A and C are not independent