Chapter 11.4, Problem 2GP

To determine

Compare the experimental and theoretical probabilities of the experiment.

Answer to Problem 2GP

Theoretical probability is greater than the Experimental probability.

Explanation of Solution

Given information:

Conduct a binomial experiment to determine the probability of drawing an even-numbered card from a 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 $20$ even-numbered cards and a total of $52$ cards.

The theoretical probability of choosing even-numbered cards from the deck of well shuffled cards is,

  $\begin{array}{l}P(\text{An even-numbered card })=\begin{array}{c}C{}_1\\ C{}_1\end{array}\\ =\begin{array}{c}20\\ 52\end{array}\\ =38.46\%\end{array}$

The theoretical probability of choosing a face card from a deck of well shuffled cards is $\boxed{38.46\%}$ .

Consider the attributes of a binomial experiment.

A trial is choosing even-numbered cards from a deck of cards and the stimulation will be of $20$ trials.

The success is choosing an even-numbered card.

The probability of success is $\begin{array}{c}5\\ 13\end{array}$ and the probability of failure is $\begin{array}{c}8\\ 13\end{array}$ .

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

Use a random number generator from $1$ to $13$ .

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

  $\begin{array}{ccc}\text{Outcome}& \text{Tally} & \text{Frequency}\\ \text{even-numbered card}& \cancel{\left|\right|\left|\right|}& 5\\ \text{Not an even-numbered card}& \cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}& 15\end{array}$

Since $5$ out of $20$ trials resulted in face card, the experimental probability of choosing an even-numbered card from a deck of cards is,

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

The experimental probability of choosing an even-numbered card from a deck of cards is $\boxed{25\%}$

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