Chapter 11.4, Problem 11PPS

To determine

The experimental and theoretical probabilities of the experiment

Answer to Problem 11PPS

  $\boxed{41.67\%}$

  $\boxed{55\%}$

Explanation of Solution

Given information:

Conduct a binomial experiment to determine the probability of pulling a red marble from a bag. 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 choosing a red marble from a set of $12$ marbles.

There are $5$ red marbles and a total of $12$ marbles.

The theoretical probability of choosing a red marble from a set of $12$ marbles is,

  $\begin{array}{l}P( \text{A red marble})=\begin{array}{c}C{}_1\\ C{}_1\end{array}\\ =\begin{array}{c}5\\ 12\end{array}\\ =41.67\%\end{array}$

Hence, the theoretical probability of choosing a red marble from a set of $12$ marbles is $\boxed{41.67\%}$ .

Consider the attributes of a binomial experiment.

A trial is choosing a red marble from a set of $12$ marbles and the stimulation will be of $20$ trials.

The success is choosing a red marble.

The probability of success is $\frac{5}{12}$ and the probability of failure is $\frac{7}{12}$ .

The random variable $X$ represents the number of red marbles drawn in $20$ trials.

Use a random number generator from $1$ to $12$ .Assign $1$ , $2$ , $3$ , $4$ and $5$ for success and the remaining $7$ marbles for failure.

The results of the experiment are as show below as per the random generator.| $\begin{array}{ccc}\text{Outcome}& \text{Tally} & \text{Frequency}\\ \text{A red marble}& \cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}|& 11\\ \text{Not a red marble}& \cancel{\left|\right|\left|\right|}\left|\right|\left|\right|& 9\end{array}$

Since $11$ out of $20$ trials resulted in a red marble, the experimental probability of choosing a red marble from a set of $12$ marbles is,

  $\begin{array}{l}\begin{array}{c}11\\ 20\end{array}=0.55\\ =55\%\end{array}$

Hence, the experimental probability of choosing a red marble from a set of marbles is $\boxed{55\%}$

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