Probability

Lesson

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.

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 {}.

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**.

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

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.

$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.

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?

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

What was the relative frequency of rolling a $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?

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

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

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