Chapter 11.4, Problem 12PPS

To determine

The experimental and theoretical probabilities of the experiment

Answer to Problem 12PPS

  $\boxed{40\%}$

  $\boxed{45\%}$

Explanation of Solution

Given information:

Conduct a binomial experiment to determine the probability of spinner stopping on an even number. 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 the spinner, numbered $1$ to $5$ stopping on an even number.

There are $2$ even numbers and a total of $5$ numbers on the spinner.

The theoretical probability of spinner stopping on an even number is,

  $\begin{array}{l}P(\text{An even number})=\begin{array}{c}C{}_1\\ C{}_1\end{array}\\ =\begin{array}{c}2\\ 5\end{array}\\ =40\%\end{array}$

Hence, the theoretical probability of spinner stopping on an even number is $\boxed{40\%}$ .

Consider the attributes of a binomial experiment.

A trial is spinning the spinner and the stimulation will be of $20$ trials.

The success is spinning landing on an even number.

The probability of success is $\begin{array}{c}2\\ 5\end{array}$ and the probability of failure is $\begin{array}{c}3\\ 5\end{array}$ .

The random variable $X$ represents the number of times that a spinner landed on an even number in $20$ trials.

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

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

  $\begin{array}{ccc}\text{Outcome}& \text{Tally} & \text{Frequency}\\ \text{An even number}& \cancel{\left|\right|\left|\right|}\left|\right|\left|\right|& 11\\ \text{Not an even number}& \cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}|& 9\end{array}$

Since $9$ out of $20$ trials resulted in an even number, the experimental probability of the spinner landing on an even number is,

  $\begin{array}{l}\begin{array}{c}9\\ 20\end{array}=0.45\\ =45\%\end{array}$

The experimental probability of spinner landing on an even number is $\boxed{45\%}$

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