Visualising probability problems on the fx-CG100

A-level maths students might find statistics and probability questions abstract and daunting, especially when faced with lots of data and multi-step reasoning.

In this blog, we explore how the fx-CG100 graphic calculator can help to demystify these problems by turning numbers into visual models, allowing easier interpretation.

Viewing probability distributions in exam questions

Let’s put the benefits of visualisation into context by looking at this question, one of the first problems posed in a past A-level statistics exam paper:

• A machine fills packets with sweets and 1/7 of the packets also contain a prize.
• The packets of sweets are placed in boxes before being delivered to shops.
• There are 40 packets of sweets in each box.
• The random variable T represents the number of packets of sweets that contain a prize in each box.
• A box is selected at random.
• Using T ~ B (40, 1/7) find:
• The probability that the box has exactly six packets containing a prize.
• The probability that the box has fewer than three packets containing a prize.

That might seem like a lot of information for students to take in when they’ve just flipped over their paper, especially if they’re feeling exam day nerves.

However, if they’ve spent enough time using the fx-CG100, they’ll know it offers all the functionality needed to work through the problem step-by-step and display the outcomes clearly.

If you were presenting this as an example in an A-level maths lesson, you could show how the Distribution app makes it easy to select the distribution type and enter the parameters before visualising the results.

To answer part a of this particular question, students should:

• Select binomial distribution
• On the Setup tab, set the tail to exact (X=)
• Change the X data value to 6
• Set the number of trials to 40
• Adjust the probability of success (P) to 1/7, which will be automatically converted to a decimal
• Press the right tab key to view the distribution on the Results tab

This will show a binomial distribution graph, and also state at the top of the screen that the probability of exactly six packets of sweets containing a prize – P (X=6) – is 0.1727.

The Tools menu allows you to move on and answer part b of this question, without returning to the Setup tab.

From the Results tab:

• Press Tools
• Select Tail
• Choose left tail (X≤) and return to the Results tab
• Highlight the X value at the top of the screen and change it to 2

This will perform a new calculation and redraw the distribution, showing that the probability of fewer than three packets containing a prize – P (X≤2) – is 0.0616.

Seeing the distribution plotted in this way can help to reinforce the idea that these aren’t just numbers, but parts of a model describing the likelihood of different outcomes. Once students can interpret the graph intuitively, they’re better prepared for the next step: using distributions to make inferences, such as testing claims or hypotheses based on sample data.

Going deeper into probability problems

The most substantial part of this exam question asks students to test the claim that the proportion of packets containing a prize is less than 1/7. They are told that nine packets contain a prize, from a random sample of 110, and asked to use a 5% level of significance.

Again, if they have the required mathematical knowledge and familiarity with the fx-CG100, the calculator provides the tools they need to unpack this question and get full marks.

In the Distribution app, students can test and visualise the following hypotheses:

They can follow the same process as outlined above to answer part a of this question, but this time choosing a left-hand tail, setting the X value to 9 and the number of trials to 110.

The Results tab will then show a highlighted portion of the probability distribution, as well as a numerical outcome that P (X ≤ 9) = 0.0383, or 3.8%. Since this is less than the 5% significance level, students can conclude there is evidence to support the claim.

By showing the left-hand tail on a graph, the fx-CG100 makes the reasoning behind the test visible. Students can see that significance isn’t about one bar of the distribution, but about how much total probability lies in the rejection region.

This link between numbers and area can aid understanding of why a small P value represents strong evidence against the null hypothesis, turning an abstract comparison into something clear and intuitive.

Exploring further with Statistics and Spreadsheet

The Distribution app is a powerful tool for visualising probabilities, but the fx-CG100 also offers other features that can support the same ideas from different angles.

In the Statistics app, for example, you can input and explore real or simulated data. Students can enter outcomes from experiments such as dice rolls, coin tosses or sampling tasks, then visualise the data in formats such as box plots, histograms and bar graphs.

This helps with linking theoretical probability models to experimental results, offering a view of concepts such as how frequencies start to stabilise around expected values.

You can also access built-in calculations like mean, standard deviation and regression to reinforce these ideas and discuss what they mean in context.

The Spreadsheet app adds another layer of flexibility, allowing you to organise and manipulate data across multiple columns. You can use it to input both values and formulas into cells, before performing statistical calculations and drawing graphs.

Together, these tools offer a range of ways to show students that probability isn’t just about abstract formulas and numbers. It can be modelled, tested and visualised directly on their calculator.

For a closer look at how the fx-CG100 supports exploration of this topic, watch our video on calculating binomial probabilities and finding critical values in hypothesis tests.

If you’re just getting started with the calculator, you can also sign up for a free introductory training session to build familiarity with its core features.