New Zealand
Level 6 - NCEA Level 1

# Experimental Probability

Lesson

## Experiment or Trial

An experiment or trial are the words used to describe the event or action of doing something and recording results. For example, the act of drawing cards from a deck, tossing a coin, rolling a dice, watching the weather, asking questions in a survey or counting cars in a carpark could all be examples of experiments or trials.

## Sample Space

The sample space, sometimes called and event space,  is a listing of all the possible outcomes that could arise from an experiment.

For example

• tossing a coin would have a sample space of {Head, Tail}, or {H,T}
• rolling a dice would have a sample space of {1,2,3,4,5,6}
• watching the weather could have a sample space of {sunny, cloudy, rainy} or {hot, cold}
• asking questions in a survey of favourite seasons could have a sample space of {Summer, Autumn, Winter, Spring}

Did you also notice how I listed the sample space? Using curly brackets {}.

## Event

An event is the word used to describe a single result of an experiment.  It helps us to identify which of the sample space outcomes we might be interested in.

For example, these are all events.

• Getting a tail when a coin is tossed.
• Rolling more than 3 when a dice is rolled
• Getting an ACE when a card is pulled from a deck

We use the notation, P(event) to describe the probability of particular events.

Adding up how many times an event occurred during an experiment gives us the frequency of that event.

The relative frequency is how often the event occurs compared to all possible events and is also known as the probability of that event occurring.

## Probability Values

The probability values that events can take on range between 0 (impossible) and 1 (certain).

## Experimental Probability

Experimental Probability, as the name suggests, describes the probability when undertaking experiments or trials.

We calculate experimental probability by considering $\frac{\text{frequency of the event }}{\text{total number of trials }}$frequency of the event total number of trials and writing it as a fraction, ratio, decimal or percentage.

#### Example

$500$500 cables were tested at a factory, and $76$76 were found to be faulty.

a) What is the experimental probability that a cable at this factory will be faulty?

$\frac{\text{frequency of the event }}{\text{total number of trials }}=\frac{76}{500}$frequency of the event total number of trials =76500

we can simplify this fraction to $\frac{19}{125}$19125, or convert it to a percentage which is $76\div500\times100=15.2%$76÷​500×100=15.2%

b) If $1500$1500 more cables were tested, how many would you expect to be faulty?

Now that we know that $15.2%$15.2% are faulty (from our experimental data), we could expect the same percentage to be faulty from any amount.

So, $15.2%$15.2% of $1500$1500

$\frac{15.2}{100}\times1500=228$15.2100×1500=228

We could expect $228$228 to be faulty from $1500$1500 cables.

#### Worked Examples

##### QUESTION 1

If the probability of an event occurring is $\frac{11}{25}$1125, how many times would you expect it to occur in $575$575 times?

##### QUESTION 2

$16$16 dice were rolled and a $2$2 occurred $4$4 times.

1. What was the relative frequency of rolling a $2$2?

2. Using the results of the trial to predict future outcomes, how many times would you expect a $2$2 to occur if $48$48 dice are rolled?

##### QUESTION 3

1000 transistors were tested at a factory, and 12 were found to be faulty.

1. What is the experimental probability that a transistor at this factory will be faulty?

2. If another 5000 transistors were tested, how many of these would you expect to be faulty?

### Outcomes

#### S6-3

Investigate situations that involve elements of chance: A comparing discrete theoretical distributions and experimental distributions, appreciating the role of sample size B calculating probabilities in discrete situations.

#### 91038

Investigate a situation involving elements of chance