Chapter 11.4, Problem 13PPS

To determine

The experimental and theoretical probabilities of the experiment

Answer to Problem 13PPS

  $\boxed{23.08\%}$

  $\boxed{20\%}$

Explanation of Solution

Given information:

Conduct a binomial experiment to determine the probability of drawing a face card out of a standard deck of cards. 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 face card from the deck of well shuffled cards.

There are $12$ face cards and a total of $52$ cards.

The theoretical probability of choosing a face card from the deck of well shuffled cards is, $\begin{array}{l}P(\text{A face card })=\begin{array}{c}C{}_1\\ C{}_1\end{array}\\ =\begin{array}{c}12\\ 52\end{array}\\ =23.08\%\end{array}$

Hence, the theoretical probability of choosing a face card from a deck of well shuffled cards is $\boxed{23.08\%}$ .

Consider the attributes of a binomial experiment.

A trial is choosing a face card from a deck of cards and the stimulation will be of $20$ trials.

The success is choosing a face card.

The probability of success is $\begin{array}{c}3\\ 13\end{array}$ and the probability of failure is $\begin{array}{c}10\\ 13\end{array}$ .

The random variable $X$ represents the number of face cards chosen in $20$ trials.

Use a random number generator from $1$ to $13$ .Assign $1$ , $2$ and $3$ for success and the remaining $10$ cards for failure.

The results of the experiment are as show below as per the random generator.|

  $\begin{array}{ccc}\text{Outcome}& \text{Tally} & \text{Frequency}\\ \text{A face card}& \left|\right|\left|\right|& 4\\ \text{Not a face card}& \cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}|& 16\end{array}$

Since $4$ out of $20$ trials resulted in face card, the experimental probability of choosing a face card from a deck of well shuffled cards is,

  $\begin{array}{l}\begin{array}{c}4\\ 20\end{array}=0.2\\ =20\%\end{array}$

Hence, the experimental probability of choosing face card from a deck of well shuffled cards is $\boxed{20\%}$

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