Chapter 5, Problem 1SE

Simulation (Example 1) If we flip a coin 10 times, how often do we get 6 or more heads? A first step to answering this question would be to simulate 10 flips. Use the random number table in Appendix A to simulate flipping a coin 10 times. Let odd digits $\left(1,3,5,7,9\right)$ represent heads, and let even digits $\left(0,2,4,6,8\right)$ represent tails. Begin with the first digit in the third row.

a. Write the sequence of 10 random digits.

b. Write the sequence of 10 "heads" and “tails." Write H for heads and T for tails.

c. How many heads did you get? Did you get 6 or more heads?

To determine

Determine the sequence of 10 digits using the random number table given in Appendix A to simulate the flip of a coin 10 times.

Answer to Problem 1SE

The sequence of 10 digits using the given random number table is given below.

$2\text{ 6 4 2 7 4 0 6 5 0}$

Explanation of Solution

The simulation is the process of collecting random outcomes for an event such that there exists the probability of observing each outcome whether it is small or large.

The theoretical probability of flipping a head is given below.

$P\left(\text{head}\right)=0.50$

Use odd digits $1,\text{ 3, 5, 7, and 9}$ to represent heads and even digits $0,\text{ 2, 4, 6, and 8}$ to represent tails. So, the probability of getting a head will be equal to the probability of getting an odd digit from the ten digits $\text{0, 1, 2, 3, 4, 5, 6, 7, 8, and 9}$.

$\begin{array}{c}P\left(\text{odd digit}\right)=\frac{5}{10}\\ =0.50\end{array}$

This probability is same as the theoretical probability of flipping a heads. Hence, this is a correct approach for simulating the probability of a heads.

Use the random number table to simulate flipping a coin. Consider all ten digits. Start from the first digit in the third row. The sequence of 10 random digits is,

$2\text{ 6 4 2 7 4 0 6 5 0}$

To determine

Determine the sequence of 10 ‘heads’ and ’tails’ to simulate the flip of a coin 10 times.

Answer to Problem 1SE

The sequence of 10 ‘heads’ and ‘tails’ is given below.

$\begin{array}{ccccccccccc}\text{Trial}& \text{1}& \text{2}& \text{3}& \text{4}& \text{5}& \text{6}& \text{7}& \text{8}& \text{9}& \text{10}\\ \begin{array}{l}\text{Outcome}\\ \text{(number)}\end{array}& \text{2}& \text{6}& \text{4}& \text{2}& \text{7}& \text{4}& \text{0}& \text{6}& \text{5}& \text{0}\\ \text{Result}& \text{T}& \text{T}& \text{T}& \text{T}& \text{H}& \text{T}& \text{T}& \text{T}& \text{H}& \text{T}\end{array}$

Explanation of Solution

Consider the sequence of 10 digits. Write H for heads representing odd digits and T for tails representing even digits.

The table given below shows the result of 10 trials using the sequence given in part(a).

$\begin{array}{ccccccccccc}\text{Trial}& \text{1}& \text{2}& \text{3}& \text{4}& \text{5}& \text{6}& \text{7}& \text{8}& \text{9}& \text{10}\\ \begin{array}{l}\text{Outcome}\\ \text{(number)}\end{array}& \text{2}& \text{6}& \text{4}& \text{2}& \text{7}& \text{4}& \text{0}& \text{6}& \text{5}& \text{0}\\ \text{Result}& \text{T}& \text{T}& \text{T}& \text{T}& \text{H}& \text{T}& \text{T}& \text{T}& \text{H}& \text{T}\end{array}$

To determine

Determine the number of heads in the flip of a coin 10 times

Answer to Problem 1SE

There are 2 heads obtained from the simulation of 10 flips.

Explanation of Solution

Consider the sequence of 10 ‘heads’ and ‘tails’ obtained by simulation.

$\begin{array}{ccccccccccc}\text{Trial}& \text{1}& \text{2}& \text{3}& \text{4}& \text{5}& \text{6}& \text{7}& \text{8}& \text{9}& \text{10}\\ \begin{array}{l}\text{Outcome}\\ \text{(number)}\end{array}& \text{2}& \text{6}& \text{4}& \text{2}& \text{7}& \text{4}& \text{0}& \text{6}& \text{5}& \text{0}\\ \text{Result}& \text{T}& \text{T}& \text{T}& \text{T}& \text{H}& \text{T}& \text{T}& \text{T}& \text{H}& \text{T}\end{array}$

From the above table, outcomes of 2 trials are heads and of remaining 8 trials are tails. So, two outcomes from the ten trials have a result of heads.

Therefore, 2 heads are obtained in the flip of a coin 10 times. 6 or more heads are not obtained.