Question

Two fair dice are rolled. Determine whether the events are dependent or independent. (a) (the first die shows 3) and (the two dice sum to 7); (b) (the first die shows 4) and (the two dice sum to 8).

Answer 1

Answer: The events are independent

a) p = 1/36 b) p = 5/216

Step-by-step explanation: both events are independent because the fact that the first die shows it sample space {1, 2, 3, 4, 5, 6} does not stop the second die from showing it own sample space too which is {1, 2, 3, 4, 5, 6}.

The sample space of first die = {1, 2, 3, 4, 5, 6}

The sample space of second die = {1, 2, 3, 4, 5, 6}

The sample space of 2 dice rolled at once is seen as attachment to this answer, find attachment below.

Probability that an event will occur = number of time the event will occur / total number of possible outcome of the event

a)

Probability that first die will show 3 = number of times 3 shows up in the event / total number of outcome of the die = P (3)

P (3) = 1/6

Probability that 2 dice sum up to give 7 = 6/ 36 = 1/6 ( from the sample space for 2 dice attached, it is seen that the sum of 7 comes out 6 times and the total sample space is 36)

P (first die show 3) = 1/6

P ( 2 dice sum to 7) = 1/6

P ( first die show 3) and P ( 2 dice sum to 7) = 1/6 * 1/6 = 1/36.

b)

Probability of the first die showing 4 = P(4) = 1/6

Probability of 2 die sum up to give 8 = 5/36

P ( first die show 8) and P ( 2 die sum up to give 8) = 1/6 * 5/36 = 5/216