Chapter 6 — Math for AI (Statistics and Probability)

Try these puzzles before reading on. What comes next in 2, 4, 6, 8, 10, 12, ...? In 4, 10, 16, 22, 28, ...? In 34, 31, 28, 25, 22, ...? A shop made a profit of ₹1000 in year 1, ₹1500 in year 2, ₹2000 in year 3 and ₹2500 in year 4. What profit do you expect in year 5?

You probably answered 14, 34, 19 and ₹3000 quickly. How? In each case you noticed a rule: add 2, add 6, subtract 3, add ₹500. An order or arrangement that repeats in a predictable way is called a pattern. Once you find the pattern, you can predict the missing or next value.

Patterns are everywhere. Patterns appear in numbers (multiplication tables, calendar dates), in images (tiles on a floor, the petals of a flower, stripes on a zebra) and in language (rhymes, grammar rules, the way 'un-' reverses a word's meaning). In picture analogy puzzles, you look at how one picture changes into the next, for example a shape rotating by a quarter turn or gaining one more dot, and use that rule to pick the missing picture.

Mathematics is the study of patterns. Arithmetic, algebra and geometry all describe regularities precisely so that we can reason about them and make predictions.

AI is a way to recognise patterns. An AI system is shown many examples and finds the patterns in them, much as you did with the number puzzles. It can find patterns in numbers (sales figures), images (the shape of a cat's ears) and speech or text (which words usually come together). These patterns let it make decisions: telling a dog from a muffin in a photo, predicting the path of a cyclone, or suggesting the next word as you type.

So AI needs mathematics. To recognise a pattern, an AI must measure, compare and calculate. Every photo is stored as a grid of numbers (pixel brightness values); every sentence is turned into numbers before a computer can process it. Mathematics is the language in which an AI describes data, learns patterns and measures how well it has learnt.

Limits of patterns. A pattern found in past data is a good guess, not a guarantee. The shop's profit may not rise by ₹500 in year 5 if a new competitor opens. AI faces the same limit: its predictions are only as good as the patterns in the data it learnt from, which is why we also study probability, the mathematics of uncertainty.

Answer these four quick questions. (1) What is the middle value of 11, 22, 33, 44, 55? (2) Of two lines drawn on a page, which is more slanted? (3) A has 2 plants, B has 3, C has 1 and D has 7: how many plants are there in all? (4) If a coin is tossed, what can the result be?

Each question uses a different branch of mathematics, and AI uses all four kinds of thinking. The CBSE course groups the mathematics of AI into four areas:

The answers to the warm-up: (1) 33 is the middle value (statistics). (2) The steeper line is the one that rises more for the same distance across; steepness is called slope (an idea from calculus). (3) 2 + 3 + 1 + 7 = 13 plants (adding many values at once, as linear algebra does with lists of numbers). (4) Heads or tails, each with probability one half (probability).

Statistics collects, organises and summarises data with measures such as the mean, median and mode, and with charts. Before an AI model is built, statistics tells us what the data looks like: its typical values, its spread and any errors.

Linear algebra works with vectors (lists of numbers) and matrices (tables of numbers) and with equations that link them. A colour photo is three huge tables of pixel values; a customer can be described by a list such as [age, visits, amount spent]. Finding an unknown from an equation, such as solving 10x - 8 = 6x to get 4x = 8 and x = 2, is the simplest version of what linear algebra does with thousands of unknowns at once.

Probability measures how likely something is, from 0 (impossible) to 1 (certain). AI systems rarely say 'definitely'; they say 'this photo is a cat with probability 0.93' or 'there is a 70% chance of rain'.

Calculus studies how one quantity changes when another changes, through ideas such as slope. When a model is trained, calculus tells it in which direction to adjust its settings so that its errors become smaller, a little at a time. You will meet calculus formally in higher classes; for now, remember that it is the tool AI uses to improve.

Do you need to be a mathematician to use AI? No. Many AI tools hide the mathematics. But understanding the basic ideas helps you ask the right questions: Is the data typical? How confident is the prediction? Could the model be wrong?

Can you find the total weight of your family members? The number of students in your school? The highest temperature in your city last month? To answer, you must collect figures, organise them and work something out. That is statistics in action.

Definition. Statistics is the branch of mathematics used for collecting, exploring and analysing data, and for drawing conclusions from it. It works in four stages:

• Collect: data is gathered from sources such as surveys, records and sensors.
• Explore and clean: the data is checked and errors, gaps and duplicates are fixed.
• Analyse: summary measures and charts are worked out to understand the data.
• Conclude: conclusions and decisions are drawn from the results.

Summary measures. A long list of numbers is hard to take in. Statistics squeezes it into a few numbers that describe it:

Example 1 (median). Heights in cm of 9 students: 155, 160, 145, 149, 150, 147, 152, 144, 148. Arrange them: 144, 145, 147, 148, 149, 150, 152, 155, 160. With 9 values, the middle one is the 5th, so the median is 149 cm. The range is 160 - 144 = 16 cm.

Example 2 (mean). A hospital records the number of patients admitted in five age groups in one day: 5, 7, 9, 22 and 11. The mean is (5 + 7 + 9 + 22 + 11) ÷ 5 = 54 ÷ 5 = 10.8 patients per group. Notice that 22 is much larger than the others; it pulls the mean up. The median of these values (5, 7, 9, 11, 22) is 9. When a dataset has an extreme value, the median often describes the 'typical' value better.

Example 3 (mode). Shoe sizes sold in a shop in one hour: 6, 7, 7, 8, 7, 9, 6. The mode is 7, so the shop should keep more size 7 in stock. The mode is the only one of the three measures that also works for qualitative data, such as the most common car colour.

Descriptive statistics. Measures like these, together with tables and charts, are called descriptive statistics, because they describe a dataset. Another important descriptive measure you will meet later is the standard deviation, which tells how far values typically lie from the mean.

In AI, these measures are used at the Data Exploration stage. A model's builders check the mean and range of every feature, look for odd values, and fill missing values, often with the mean or median of the column.

Statistics is not just a chapter in a textbook. Governments, hospitals, sports teams and weather offices make big decisions every day from statistical summaries of data. The CBSE course highlights four areas.

1. Disaster management. Authorities study records of past floods, cyclones and earthquakes: where they struck, how strong they were and how often. Combined with live data from satellites and rain gauges, this tells them which areas are at risk, so they can warn residents and plan evacuations. Statistics on population, hospitals, roads and shelters in an area tells relief teams how many people may need help and where to send food, boats and medical teams. India's improved cyclone warnings, which have helped move large numbers of people to safety in recent years, depend heavily on such data.

2. Sports. Teams record every ball, pass and shot. A batter's average, strike rate and scoring areas help a captain set the field; a bowler's economy rate decides when to bring him on. Statistics also guides decisions off the field. When the Tokyo 2020 Olympics were planned, data on the sharp rise of COVID-19 cases was a key reason the Games were postponed to 2021.

3. Disease prediction and public health. Health departments count cases of diseases by district and by week. If dengue cases in one ward double in a fortnight, that is a signal to send fogging teams there. During COVID-19, statistics showed where cases were rising and where vaccination needed to speed up. Hospitals also use statistics to find which diseases affect the most people, so that resources go where they help most.

4. Weather forecasting. Forecasters compare today's measurements of temperature, pressure, humidity and wind with records of past seasons. Computer models, built on statistics and physics, then estimate tomorrow's weather. Farmers, airlines and event organisers plan around these forecasts.

Statistics about you. Surveys of young people collect statistics on screen time, sleep, exercise and school attendance. For example, UNESCO estimates that hundreds of millions of children and young people around the world are out of school. Such statistics help governments design schemes and measure whether they are working. When you read such figures, apply your data literacy: who collected them, when, and from how many people?

Where AI fits. In all four areas, AI systems are now trained on the same statistical data to find patterns faster than people can: predicting a cyclone's path, spotting the start of an outbreak, or suggesting a batting order. The CBSE activity Uses of Statistics in Daily Life asks you to collect a small dataset of your own, such as the time your family spends on screens each day, and apply the mean, median and mode to draw a conclusion.

In the CBSE activity Car Spotting and Tabulating, you watch a video of a road (or stand safely near your school gate) for a fixed time and record the colour of every car that passes. It is a small project, but it uses the full chain of statistics: data collection, organisation, analysis and interpretation, which is also the chain that every AI system depends on.

Step 1: Collect with tally marks. Write the colours in a column. Each time a car passes, draw a stroke next to its colour. Every fifth stroke is drawn across the previous four, so the marks fall into easy-to-count bundles of five. Tallies are fast to record and hard to lose count with.

Step 2: Organise into a frequency table. Count the strokes for each colour. The count is the frequency. Suppose in 10 minutes you recorded:

(In real tallies the fifth stroke crosses the first four; the table above simply groups them in fives.)

Step 3: Analyse.

• How many cars were spotted in total? 9 + 6 + 5 + 3 + 2 = 25.
• Which colour was spotted most often? White, with frequency 9. This is the mode of the data.
• What fraction of cars were white? 9 out of 25 = 0.36 = 36%.
• How many more white cars than red ones? 9 - 3 = 6.

Step 4: Present. A bar graph with one bar per colour makes the comparison clear at a glance (see the figure). Colour is qualitative data, so a bar graph, with a gap between bars, is the right choice.

Step 5: Interpret. The most common colour choice among people who drive on this road seems to be white. But a data literate student adds cautions: the sample is only 25 cars from one road in 10 minutes; a different time of day, a different road or a bigger sample might give different results. A good conclusion states its limits.

Why this matters for AI. An AI that counts vehicles from traffic cameras does exactly these steps automatically: it detects each car (computer vision), records its type and colour (data collection), counts frequencies (statistics) and reports patterns such as peak hours (interpretation). The quality of its conclusions still depends on whether its data is large and representative enough, the same caution you applied above.

Try it yourself. Repeat the activity for a different feature, such as the type of vehicle (car, two-wheeler, bus, auto) or the number of people in each car, and compare the frequency tables of two different times of day.

Before a cricket match, the captains toss a coin. Nobody knows whether it will land heads or tails, but everyone agrees that each is equally likely. Probability is the branch of mathematics that measures how likely something is to happen.

Words we need.

• An experiment is an action with uncertain results, such as tossing a coin or rolling a die.
• An outcome is one possible result: heads, or a 4 on the die.
• An event is an outcome or group of outcomes we are interested in, such as 'getting an even number'.
• Favourable outcomes are the outcomes that make the event happen.

The formula. When all outcomes are equally likely,

Probability of an event = number of favourable outcomes ÷ total number of possible outcomes

Coin. A toss has 2 outcomes, H and T. For heads, 1 outcome is favourable, so P(heads) = 1/2 = 0.5. Likewise P(tails) = 1/2.

Die. A standard die has 6 faces: 1, 2, 3, 4, 5, 6. P(getting 4) = 1/6. The event 'even number' has the favourable outcomes 2, 4 and 6, so P(even) = 3/6 = 1/2. P(number greater than 4) = 2/6 = 1/3, since only 5 and 6 qualify.

Bag of balls. A bag has 7 red and 3 blue balls. A ball is drawn without looking. There are 10 possible outcomes. P(red) = 7/10 = 0.7 and P(blue) = 3/10 = 0.3.

Key facts.

• Every probability lies between 0 and 1. It can also be written as a percentage from 0% to 100%.
• A probability of 0 means the event is impossible; 1 means it is certain.
• The probabilities of all the possible outcomes add up to 1: P(red) + P(blue) = 0.7 + 0.3 = 1.
• So P(event does not happen) = 1 - P(event happens). If P(rain) = 0.7, then P(no rain) = 0.3.

Probability is about the long run. P(heads) = 1/2 does not mean that in 10 tosses you will get exactly 5 heads. You might get 3 or 7. But over thousands of tosses the fraction of heads gets close to 1/2. This idea of long-run frequency is also how probabilities are estimated from data when outcomes are not equally likely: if it rained on 120 of the last 400 July days, the estimated probability of rain on a July day is 120/400 = 0.3.

Probability in AI. AI systems express their answers as probabilities. An image classifier may output: cat 0.92, dog 0.07, rabbit 0.01, which add up to 1. It reports 'cat' because that has the highest probability, and the 0.92 shows how confident it is.

Think about these: the sun rising in the east tomorrow; rolling a 7 on an ordinary die; getting heads on a coin; picking a red ball from a bag with 7 red and 3 blue. Some are sure to happen, some can never happen, and some lie in between. Describing how likely an event is in words is the first step in probability.

The probability scale. These words sit along a line from 0 to 1 (see the figure). Impossible is at 0, certain at 1, and an even chance at the middle, 0.5. Events near 1 are likely; those near 0 are unlikely.

The bag of stars. A bag holds 7 yellow stars and 3 green stars. Without looking, you pick one.

• It is likely that you pick a yellow star (7/10 = 0.7).
• It is unlikely that you pick a green star (3/10 = 0.3).
• It is impossible to pick a red star (0/10 = 0), as there are none.
• If you remove 4 yellow stars, 3 yellow and 3 green remain, so there is an equal probability of picking yellow or green (3/6 each).
• You will certainly pick a star (10/10 = 1), since everything in the bag is a star.

A relatable example. You want to go to a friend's party, and your parents make a deal. If they say 'you can go if pigs fly', your chance is impossible. If they say 'you can go if you finish your homework', and you always do, it is certain. If they say 'we will toss a coin', it is an even chance. If it depends on getting full marks in tomorrow's test, it may be unlikely; if it depends on simply tidying your room, it is likely.

Classify the CBSE exercise events. Tossing a coin and getting heads: equal probability (1/2). Rolling an 8 on a standard die: impossible. Throwing ten 5s in a row: possible but very unlikely, since the probability is (1/6) multiplied by itself ten times, which is about 0.000000017. Drawing a card of some suit from a full pack: certain, since every card has a suit.

Why AI cares. An AI model must know the difference between 'unlikely' and 'impossible'. A fraud-detection system flags transactions that are very unlikely for that customer, such as a large purchase in another country at 3 a.m., without treating them as impossible.

You use probability every day without writing any fractions. You carry an umbrella because rain looks likely; you leave early because traffic is usually heavy at 6 p.m. The CBSE course highlights three areas where probability is applied, and each is also a place where AI is now used.

1. Sports. Teams estimate probabilities from past performance. Suppose a batter scored 45 of her last 100 runs in boundaries. A simple estimate is that about 45% of her runs in the next match will also come from boundaries, so the captain may post fielders on the boundary. Win-probability charts on cricket broadcasts work the same way: from thousands of past matches, they estimate the chance that a team chasing a given target with a given number of wickets and balls left will win.

2. Weather forecasting. A forecast of '70% chance of rain between 4 p.m. and 6 p.m.' means that, on days with similar conditions in the past, it rained in that period about 7 times out of 10. It gives a medium-to-high likelihood, not a promise. So P(no rain) = 1 - 0.7 = 0.3: three similar days in ten stay dry. Farmers decide when to sow and spray, and cities prepare drains, using such probabilities.

3. Traffic estimation. From experience of the time of day, the area of the city and the weather, people judge how bad traffic is likely to be. If you think there is a 90% chance that traffic near you will be heavy between 6 p.m. and 7:30 p.m., you might wait until later to leave. Navigation apps do this on a large scale: they combine live speeds from many phones with historical patterns to estimate the probable travel time on each route and suggest the fastest one.

More examples.

• Insurance: companies set premiums from the probability of accidents or illness in each age group.
• Medicine: doctors weigh the probability that a test result is correct and that a treatment will work.
• Games: in Ludo, the chance of rolling the 6 you need is 1/6 on each throw.
• Quality control: a factory estimates the probability that an item is faulty by testing a sample.

Probability inside AI. Most AI systems work with probabilities rather than certainties:

A caution. A probability is not a promise. A 90% chance of a sunny day still leaves a 10% chance of rain, and on some days that is what happens. Good decisions use probability to weigh risks: carrying an umbrella costs little, so even a 30% chance of rain may justify it, while an important decision may need a much higher probability before you act.