Chapter 11.4, Problem 4CYU

To determine

Find the theoretical and experimental probability.

Answer to Problem 4CYU

Theoretical probability $\boxed{15.38\%}$ , experimental probability $\boxed{25\%}$ .

Explanation of Solution

Given information:

Conduct a binomial experiment to determine the probability of drawing an ace or a king from a deck of cards. Then compare the experimental and theoretical probabilities of the experiment.

Calculation:

Generally, the probablity of event $E$ is,

  $P(E)=\frac{Number-of-favorable-outcomes}{Total-number-of-outcomes}$

Consider the probability of choosing an ace or a king a deck of well shuffled cards.

There are $4$ aces, $4$ kings and a total of $52$ cards.

The theoretical probability of choosing an ace or a king from a deck of well shuffled cards is,

  $P$ ( An ace or a king)= $\frac{{}^4{C}_1{+}^4{C}_1}{{}^{52}{C}_1}$

  $\begin{array}{l}=\frac{4+4}{52}\\ =\frac{8}{52}\\ =15.38\%\end{array}$

Hence, the theoretical probability of choosing an ace or a king from a deck of well shuffled cards is $\boxed{15.38\%}$

Examine the following attributes of a binomial experminet.

A trial os choosing a card at random from a deck of cards and the stimulation will be of $20$ trials.

The success is choosing an ace or a king.

The probability of success is $\frac{2}{13}$ and the probability of failure is $\frac{11}{13}$ .

The random variable $X$ represents the number of aces or king chosen in $20$ trials.

Use a random number generator from $1$ through $13$ . Assign $1$ and $2$ for success and the remaining $11$ cards for failure.

The results of the experiment are as shown below as the random generator.

  $\begin{array}{ccc}0utcome& Tally& Frequency\\ An-ace-or-a-king& \cancel{\left|\right|\left|\right|}& 5\\ Not-an-ace-or-a-king& \cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}& 15\end{array}$

Since $5$ out of $20$ trials resulted in an ace or a king, the experimental probability of choosing an ace or a king from a deck of well shuffledcards is,

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

Hence, the experimental probability of choosing an ace or a king from a deck of well shuffled cards is $\boxed{25\%}$ .

When we compare both of them, it is clear that, the experimental probability is greater than the theoretical probability.