Eduki

Interpreting the meaning of probability - what does it mean by a probability of 1/2?

Let’s look at our modified spinner, the one that we used earlier. We calculated the probability of the event “blue” to be 2/4 or ½.

[spinner here]

But what does it mean? Does it mean that if we spin the spinner 2 times, we’ll get the event “blue” one time, or that if we spin the spinner 6 times, we’ll get the event “blue” 3 times? Will we always get the “blue” event half of the time?

Let’s do an experiment to see what happens. Let’s spin the spinner 1000 times and count the number of “blue” events. This is very tedious, let’s ask the computer to do this. Here are the results.

[table here]

You can see that as we repeat the experiment, the estimate of the probability for the event “blue” gets close to what is expected from the exact probability in the long run; however, the outcome for a chance event is not guaranteed and estimates of the probability for an event using short-term results will not usually match the actual probability exactly.  

Probability describes what happens in the long run and it does not guarantee that the event will occur a specific number of times after any specific number of trials. An event that has a probability of 1/2 means that the event will occur about 50% of the time in the long run, but it does not mean that it will occur exactly 50 times when the experiment is performed 100 times. So, a probability for an event represents the proportion of the time we expect that event to occur in the long run.

We can also see that the fraction of outcomes for which the event occurs, in the long run, gives an estimate of the probability of the event occurring. Probability computed this way is called empirical probability and it is especially useful when we do not know the sample space to calculate probability analytically. We will see an example of this a bit later. 

If we get three tails in a row, are we more likely to get a head?

A probability for an event represents the proportion of the time we expect that event to occur in the long run. For example, the probability of a coin landing heads up after a flip is, which means that if we flip a coin many times, we expect that it will land heads up about half of the time.

Even though the probability tells us what we should expect if we flip a coin many times, that doesn't mean we are more likely to get heads if we just got three tails in a row. The chances of getting heads are the same every time we flip the coin, no matter what the outcome was for past flips.  

What does it mean when the fraction of the time an event occurs may differ from the actual probability?

There are two main reasons why the fraction of the time an event occurs may differ from the actual probability:

1. The experiment or simulation was designed or run poorly.
2. Not enough trials were run.

So, let's say you roll a die 10 times and you got "3" eight times? What might have happened?

Well, 10 trials are not enough; we could get "3" eight times just by chance. If you run the trial 600 or 6000 times, it is more likely that you will get close to 100 or 1000 "3"s.

In another scenario, you did roll a die 1000 times, but you got only 70 threes. What could have happened?

In this case, the number is trials is quite big. So, I would suspect the die is not fair, i.e. there might be some uneven surface or corners in the die such that there is a lower likelihood of getting three when the die is rolled.  

Estimating probability when we do not know the entire sample space

So far, we have only dealt with situations where all sample spaces were obvious before actually doing the experiment. For instance, in the past, we could see that the spinner has four possible outcomes and therefore the sample space of four. Let’s have a look at the new spinner. Here, we do not know the exact sample space. All we know is that the blue area is much bigger than the green and red sections, but we don’t know how much exactly. 

We could approach such problems by estimating the probability of an event using the results from repeating trials. We can spin the wheel many times and note down the outcomes. We could then calculate the probability of the event “blue” by finding the fraction of the outcome that resulted in “blue”.

Let’s do that. Below are the observations:

# trials	# blue	# red	# green
100
1000

So, we could estimate the probability of the event “blue” to be x. 

If I told you that this is very close to the actual probability of “blue”, as you can see in the figure below.

[figure below with sections]

Well, in this case at least we could see the sample space, even though we did not know the exact sample space. In real life, we come across many situations where we do not know the sample space. Let’s look at an example. 

We have a bag of blocks that are different colors. We can’t look into the bag, so we don’t know the sample space. What do you think is the probability of taking out a green block from this bag? 

Just like before, we can randomly draw an item from the bag, record it and put the block back and shuffle. Let’s make sure we don’t look into the bag and take only one block out at a time. We can repeatedly draw items from the bag and look for patterns in this repetition. 

# trials	# blue	# red	# green
20	3	5	12

Because there are 12 out of 20 outcomes were green, we could estimate that the probability of taking out a green block from this bag is 13/20. 

Could we get a better estimate without opening the bag?

Yes, if we repeated the experiment more times, our estimates will get better. Let’s do it 300 times. 

# trials	# blue	# red	# green
20	2	5	13
200	30	50	120

Well, after repeating the trial 200 times, we calculate the probability of taking out a green block as 12/20 or ⅗. This is still an estimate, but this is a better estimate as it uses more repeated trials. If you need to find the exact probability, you will need to open the bag and count the exact number of blocks.