Question

Die and coin. Consider the following random experiment : First , roll a die and observe the number of dots facing up: then toss a coin the number of times that the die shows and observe the total number of heads. Thus , if the die shows three dots facing up and the coin (which is then tossed tree times) comes up heads exactly twice, then the outcome of the experiment can be represent as (3,2).

Part (a) Determine a sample space for this experiment.

Part (b) Determine the events that the total number of heads is even.

Short answer

Part (a).

$$\begin{gathered} U=\{\left(1,0\right),\left(1,1\right),\left(2,0\right),\left(2,1\right),\left(2,2\right),\left(3,0\right),\left(3,1\right),(3,2),\left(3,3\right), \\ \left(4,0\right),\left(4,1\right),\left(4,2\right),(4,3),\left(4,4\right),(5,0),\left(5,1\right),\left(5,2\right),\left(5,3\right),\left(5,4\right), \\ (5,5),(6,0),\left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),(6,6\left)\right\} \end{gathered}$$

Part (b)$E=\{\left(1,0\right),\left(2,0\right),\left(2,2\right),\left(3,0\right),\left(3,2\right),\left(4,0\right),\left(4,2\right),\left(4,4\right),\left(5,0\right),\left(5,2\right),\left(5,4\right),\left(6,0\right),\left(6,2\right),\left(6,4\right),(6,6\left)\right\}$

Step-by-step solution

Part (a) Step 1. Given information.

When a die is rolled, the number of dots (n) facing up is recorded. Then a coin is tossed that many times (n) and the number of heads (k) that appear is counted. If a die is tossed and it comes up six, the coin will be tossed six times and the head will appear four times. The experiment's result can then be represented as (6, 4).

Part (a) Step 2. All possible outcomes are represented in the sample space: 

Possible outcome:
$$\begin{gathered} U=\{\left(1,0\right),\left(1,1\right),\left(2,0\right),\left(2,1\right),\left(2,2\right),\left(3,0\right),\left(3,1\right),(3,2),\left(3,3\right), \\ \left(4,0\right),\left(4,1\right),\left(4,2\right),(4,3),\left(4,4\right),(5,0),\left(5,1\right),\left(5,2\right),\left(5,3\right),\left(5,4\right), \\ (5,5),(6,0),\left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),(6,6\left)\right\} \end{gathered}$$

Part (b) Step 1. Given information.

When a die is rolled, the number of dots (n) facing up is recorded. Then a coin is tossed that many times (n) and the number of heads (k) that appear is counted. If a die is tossed and it comes up six, the coin will be tossed six times and the head will appear four times. The experiment's result can then be represented as (6, 4).

Part (b) Step 2. The event in which the total number of heads is even:

Possible outcome:

$E=\{\left(1,0\right),\left(2,0\right),\left(2,2\right),\left(3,0\right),\left(3,2\right),\left(4,0\right),\left(4,2\right),\left(4,4\right),\left(5,0\right),\left(5,2\right),\left(5,4\right),\left(6,0\right),\left(6,2\right),\left(6,4\right),(6,6\left)\right\}$