10.015 Experimental probability

Experimental probability

As we mentioned before, an event is a set of outcomes for a random experiment. For example, flipping a coin and getting heads, tossing a die and getting an even number. An experiment is a repeatable procedure with a set of possible outcomes. For example, tossing a die, flipping a coin. Experimental probability, as the name suggests, describes the probability of an event occurring when undertaking experiments or trials.  It is also sometimes referred to as relative frequency.

It can be calculated as follows:

$\text{Experimental Probability}=\frac{\text{Number of favourable outcomes}}{\text{Total number of trials}}$Experimental Probability=Number of favourable outcomesTotal number of trials​

What actually happens may or may not be the same as what you expect using theoretical probability. For example, we all know that there is a 50% chance of landing a tails when you flip a coin. But it is possible that after 10 flips, you'll collect 7 tails and only 3 heads! The longer an experiment continues, the more likely it is to result in experimental probabilities that are close to the theoretical probability.

The graph below shows the results of tossing a coin 50 times. As we can see the probability varies significantly at the start but as we increase the number of coin tosses the probability settles and approaches the theoretical probability of $0.5$0.5.

 

Experimental probability is very important in cases where the theoretical probability cannot be calculated. Such as research or experiments in environmental sciences, behavioural sciences, economics, medicine and marketing. For example: if we want to know the probability of catching a fish over a certain size from a large lake, how could we find the probability? We couldn't know the theoretical probability unless we caught and measured every fish in the lake. However, working from previous data of sizes of fish caught in the lake we can use the experimental probability for an estimate of the actual probability. 

We can use experimental probabilities as point estimates for the actual probability. For our estimate to be reasonable the number of trials must be sufficiently large. There are many other experimental design factors that come into play to ensure the sample is representative of the population. 

We can also use experimental probabilities to test systems that we know the theoretical probability of. For example, we could test to see if a coin is biased. For a small number of trials we would expect the probability to vary, but if we tossed the coin many times we would expect the experimental probability to be close to $0.5$0.5. So if we tossed a coin $100$100 times and had an experimental probability of $0.3$0.3 for obtaining a tail, we could conclude with reasonable certainty that the coin is biased (this means it favours one outcome over the others). 

Worked example

Example 1

$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?

$P\left(\text{Faulty}\right)$P(Faulty)	$=$=	$\frac{\text{Number of favourable outcomes}}{\text{Total number of trials}}$Number of favourable outcomesTotal number of trials​
	$=$=	$\frac{76}{500}$76500​
	$=$=	$0.152$0.152

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

Now that we know that approximately $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$1500$=$=$0.152\times1500=228$0.152×1500=228

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

Practice questions

Question 1

Question 2