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# Theoretical and Experimental Probability

## Ratio of successes to sum of successes and failures

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Basic Probability - Probability and Statistics

Most people have heard, I think, of the old adage that buttered bread always lands buttered side down. However, from a scientific standpoint, what is the real statistical and experimental probability of buttered bread landing butter side up? For that matter, what is the difference between a statistical and an experimental probability?

Credit: Jarkko Laine
Source: https://www.flickr.com/photos/jarkkolaine/8184789759
License: CC BY-NC 3.0

### Basic Probability

Probability is the study of chance. When studying probability, there are two very general classifications: theoretical probability and experimental probability.

• Theoretical probability is the calculated probability that a given outcome will occur if the same experiment were completed an infinite number of times.
• Experimental probability is the observed result of an experiment conducted a limited number of times.

For example, ignoring the very slight differences between the figures stamped onto each side of a coin, the statistical probability of a coin landing heads-up is 50%. However, if you flip a coin 10 times, you may very well find that the observed experimental probability results in 60% or 70% or even greater probability of one side landing up. This discrepancy is perfectly natural and expected when conducting experiments, and it is important to recognize it.

In this lesson we will confine our study to the probability of a simple event. The probability of a simple event is the calculated chance of a specific direct outcome of a single experiment where in all possible outcomes are equally likely. To calculate the probability of such an outcome, we use a very simple and intuitive formula:

\begin{align*}P(x)=\frac{\text{number of events where} \ x \ \text{is true}}{\text{total number of possible events}}\end{align*}

Where “\begin{align*}P(x)\end{align*}” is the probability that \begin{align*}x\end{align*} will occur

In other words, just as you might expect, the probability of randomly picking one of the three blue marbles out of a bag with ten marbles total would be \begin{align*}\frac{3}{10}\end{align*}

#### Calculating Probability

1. You are given a big containing 15 equally sized marbles. You know there are 10 yellow marbles and 5 green marbles in the bag. What is the probability that you would pull a yellow marble out, if you reach in the bag and grab a marble at random?

Use the formula for the probability of a simple event:

\begin{align*}P(x)=\frac{\text{number of outcomes where} \ x \ \text{is true}}{\text{total number of possible outcomes}}\end{align*}

In this case, we have:

\begin{align*}P(yellow)=\frac{10 \ yellow \ marbles}{15 \ total \ marbles}\end{align*}

Which would reduce to:

\begin{align*}P(yellow)=\frac{2}{3} \ or \ 66.6 \bar{6} \% \end{align*}

2. What is the probability of rolling an odd number on a standard six-sided die?

A standard die has three odd numbers (1, 3, 5) and three even numbers (2, 4, 6). Therefore, the probability of rolling an odd number is:

\begin{align*}P(odd)=\frac{3 \ odd}{6 \ total}\end{align*}

Reducing to:

\begin{align*}P(odd)=\frac{1}{2} \ or \ 50 \%\end{align*}

Finding Experimental Probability

If Lawrence is playing with a standard 52-card deck, then the statistical probability of him pulling a single Queen at random is: \begin{align*}\frac{4 \ queens}{52 \ cards}=\frac{1}{13}=7.7 \%\end{align*}. If he decides to test it out and ends up pulling a Queen at random 6 times in 52 trials of “pull a card, record it, put it back”, what is the experimental probability of pulling a Queen?

Recall that experimental probability is the observed probability of a number of identical experiments. Experimental probability is not affected by statistical probability (it may be predicted by it, but not affected), therefore the experimental probability is:

\begin{align*}P(y)=\frac{6 \ Queens}{52 \ trials}\end{align*}

Reducing to:

#### Earlier Problem Revisited

From a scientific standpoint, what is the real statistical and experimental probability of buttered bread landing butter side up? For that matter, what is the difference between a statistical and an experimental probability?

Remember that the difference is that statistical probability is the calculated probability of a specific outcome, and experimental probability is the observed probability.

The statistical probability of the bread landing butter side up can be assumed to be \begin{align*}\frac{1}{2}\end{align*}, based on bread having two sides.

According to the “MythBusters” experiment in the video, the observed probability was \begin{align*}\frac{29}{45}\end{align*}. However, you should know that your results might be different!

### Examples

#### Example 1

What is the probability of pulling the 1 red marble out of a bag with 12 marbles in it?

\begin{align*}P(red)=\frac{\text{1 red marble}}{\text{12 total marbles}}=\frac{1}{12} \ or \ 8.3 \%\end{align*}

#### Example 2

What is the probability of a spinner landing on “6” if there are 6 equally spaced points on the spinner?

\begin{align*}P(red)=\frac{\text{1 red marble}}{\text{12 total marbles}}=\frac{1}{12} \ or \ 8.3 \%\end{align*}

#### Example 3

What is the probability of pulling a red card at random from a standard deck?

\begin{align*}P(red)=\frac{\text{26 red cards}}{\text{52 total cards}}=\frac{26}{52}=\frac{1}{2} \ or \ 50 \%\end{align*}

#### Example 4

What is the experimental probability of heads in an experiment where Scott flipped a coin 50 times and got heads 21 times?

\begin{align*}P(heads)=\frac{\text{21 heads}}{\text{50 flips}}=\frac{21}{50} \ or \ 42 \%\end{align*}

#### Example 5

What is the probability of shaking the hand of a female student if you randomly shake the hand of one person in a room with 23 female students and 34 male students?

\begin{align*}P(female)=\frac{\text{23 females}}{\text{57 students }}=\frac{23}{57} \ or \ 40.4 \%\end{align*}

### Review

Questions 1-10, find the probability:

1. Rolling a 4 on a standard die
2. Pulling a King from a standard deck
3. Pulling a green candy from an opaque bag with 5 red, 3 yellow, 3 blue, and 6 green candies.
4. Getting a 5 from one spin on a spinner numbered 1-8 (equally spaced)
5. Rolling an even number on a 20-sided die
6. Rolling and odd number on a standard die
7. Pulling a red card from a standard deck
8. Pulling a face card from a standard deck
9. Spinning red on a spinner with Red, Orange, Yellow, Green, Blue and Purple (equally spaced)
10. Pulling a club from a standard deck
11. Pulling a brown candy from a box of 25 candies, containing equal numbers of brown, red, green, blue, and yellow candies
12. Getting a prime number with a random number generator that has an equal chance of generating any number between 1 and 50
13. Getting a composite number with the same generator

To view the Review answers, open this PDF file and look for section 6.1.

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### Vocabulary Language: English

Event

An event is a set of one or more possible results of a probability experiment.

experimental probability

Experimental (empirical) probability is the actual probability of an event resulting from an experiment.

Outcome

An outcome of a probability experiment is one possible end result.

theoretical probability

Theoretical probability is the probability ration of the number of favourable outcomes divided by the number of possible outcomes.

trial

A trial is one “run” of a particular experiment.