Answer to Question #167150 in Statistics and Probability for mandie sun

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