Chapter 11.4, Problem 10PPS

To determine

The experimental and theoretical probabilities of the experiment

Answer to Problem 10PPS

  $\boxed{16.67\%}$

  $\boxed{25\%}$

Explanation of Solution

Given information:

Conduct a binomial experiment to determine the probability of rolling a $7$ with two dice. Then compare the experimental and theoretical probabilities of the experiment.

Calculation:

In general, the probability of event $E$ is,

  $P(E)=\begin{array}{c}\text{Number of favourable outcomes}\\ \text{Total number of outcomes}\end{array}$

Consider the probability of rolling a $7$ with two dice .

There are $6$ outcomes in which a pair dice rolls $7$ and a total of $36$ outcomes.

The theoretical probability of rolling a $7$ with two dice is,

  $\begin{array}{l}P( \text{rolling a 7})=\begin{array}{c}6\\ 36\end{array}\\ =\begin{array}{c}1\\ 6\end{array}\\ =16.67\%\end{array}$

Hence, the theoretical probability of rolling a $7$ with two dice is $\boxed{16.67\%}$ .

Consider the attributes of a binomial experiment.

A trial is rolling two dice and the stimulation will be of $20$ trials.

The success is rolling a $7$ .

The probability of success is $\frac{1}{6}$ and the probability of failure is $\frac{5}{6}$ .

The random variable $X$ represents the number $7$ ’s rolled in $20$ trials.

Use a random number generator from $1$ to $6$ .Assign $1$ for success and the remaining $5$ for failure.

The results of the experiment are as show below as per the random generator.| $\begin{array}{ccc}\text{Outcome}& \text{Tally} & \text{Frequency}\\ \text{ rolling a 7}& \cancel{\left|\right|\left|\right|}& 5\\ \text{rolling a number other than 7}& \cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}& 15\end{array}$

Since $5$ out of $20$ trials resulted in an ace or king, the experimental probability of rolling a $7$ with two dice is,

  $\begin{array}{l}\begin{array}{c}5\\ 20\end{array}=0.25\\ =25\%\end{array}$

Hence, the experimental probability of choosing a red marble from a set of marbles is $\boxed{25\%}$

When compared, it is clear that the experimental probability is greater than the theoretical probability.