Limits, Probability, and Statistics: math lesson, Grade 12

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Math lessons Grade 12 : Limits, Probability, and Statistics — Zyro the alien explorer of Planète Maths

Some questions ask what a quantity approaches: the speed of a car at one exact instant, or the cost per item when a factory makes huge batches. Others ask how likely an event is, or how to summarize a pile of data. This chapter joins both worlds. You will study limits, count arrangements, compute probabilities with the binomial and normal models, and fit a line to real data.

Sommaire

1. Intuitive limits and one-sided limits

Limit

We write \( \lim_{x\to a} f(x) = L \) when the values \( f(x) \) get as close as we like to \( L \) as \( x \) gets close to \( a \) from both sides, with \( x \neq a \). The value \( f(a) \) itself does not matter, and it may not even exist.

Example 1: a hole in the graph

Let \( f(x) = \dfrac{x^2-9}{x-3} \). We cannot plug in \( x=3 \) because that gives \( \tfrac{0}{0} \). Try nearby values: \( f(2.9)=5.9 \), \( f(2.99)=5.99 \), \( f(3.01)=6.01 \), \( f(3.1)=6.1 \). The outputs close in on 6. Algebra confirms it: for \( x\neq 3 \), \( f(x)=\dfrac{(x-3)(x+3)}{x-3}=x+3 \), so \( \lim_{x\to 3} f(x)=6 \).

Finding a limit at a number

Try direct substitution. If you get a real number, that is the limit (for a polynomial or a nice rational function).
If you get \( \tfrac{0}{0} \), factor, simplify, or multiply by a conjugate, then substitute again.
If the function changes formula at \( a \), look at each side separately.

One-sided limits

\( \lim_{x\to a^-} f(x) \) is the value approached when \( x \) comes from the left (\( x \lt a \)). \( \lim_{x\to a^+} f(x) \) is the value approached from the right (\( x \gt a \)).

Two sides must agree

\( \lim_{x\to a} f(x) \) exists if and only if both one-sided limits exist and are equal.

Example 2: a jump

Let \( f(x)=\begin{cases} 2x-1 & \text{if } x\lt 1\\ x^2+2 & \text{if } x\ge 1\end{cases} \). From the left, \( \lim_{x\to1^-} f(x)=2(1)-1=1 \). From the right, \( \lim_{x\to1^+} f(x)=1^2+2=3 \). Since \( 1\neq 3 \), the limit at \( x=1 \) does not exist, even though \( f(1)=3 \).

2. Limits at infinity

Here we ask what happens to \( f(x) \) as \( x \) grows without bound. We write \( \lim_{x\to\infty} f(x)=L \) when the graph levels off at the height \( L \); the line \( y=L \) is a horizontal asymptote. Remember that \( \tfrac{1}{x} \), \( \tfrac{1}{x^2} \), and similar terms shrink to 0.

Rational functions at infinity

For \( f(x)=\dfrac{a_n x^n+\cdots}{b_m x^m+\cdots} \): if \( n\lt m \) the limit is 0; if \( n=m \) the limit is \( \dfrac{a_n}{b_m} \); if \( n\gt m \) there is no finite limit (the values grow without bound).

Example 3: dividing by the highest power

For \( \lim_{x\to\infty}\dfrac{6x^2-5x}{2x^2+7} \), divide every term by \( x^2 \): \( \dfrac{6-\frac{5}{x}}{2+\frac{7}{x^2}}\to \dfrac{6}{2}=3 \). The line \( y=3 \) is a horizontal asymptote.

The graph above shows \( f(x)=\dfrac{2x+1}{x-1} \). Its horizontal asymptote is \( y=2 \) because the degrees match and \( \tfrac{2}{1}=2 \). It also has a vertical asymptote at \( x=1 \), where the denominator is 0 and the numerator is not.

Careful

\( \infty \) is not a number. Writing \( \lim_{x\to\infty} x^2=\infty \) only describes how the values behave; it does not mean the limit exists as a real number.

3. Permutations and combinations

To count outcomes, use the fundamental counting principle: if one choice can be made in \( m \) ways and a second in \( n \) ways, the two together can be made in \( m\cdot n \) ways. The factorial is \( n!=n(n-1)\cdots 2\cdot 1 \), with \( 0!=1 \).

Permutations and combinations

A permutation is an arrangement where order matters: \( P(n,r)=\dfrac{n!}{(n-r)!} \). A combination is a selection where order does not matter: \( C(n,r)=\dfrac{n!}{r!\,(n-r)!} \). They are linked by \( P(n,r)=r!\cdot C(n,r) \).

Example 4: medals or committees

Eight sprinters race for gold, silver, and bronze. Order matters: \( P(8,3)=8\cdot7\cdot6=336 \) podiums. A coach picks 4 of 10 students for a committee. Order does not matter: \( C(10,4)=\dfrac{10\cdot9\cdot8\cdot7}{4!}=210 \).

Example 5: repeated letters

The word LEVEL has 5 letters, with L twice and E twice. The number of distinct arrangements is \( \dfrac{5!}{2!\,2!}=\dfrac{120}{4}=30 \).

Order test

Zyro asks: "If I swap two chosen items, do I get a different outcome?" If yes, use a permutation. If no, use a combination.

4. Conditional probability

Conditional probability

The probability of \( A \) given that \( B \) has occurred is \( P(A\mid B)=\dfrac{P(A\cap B)}{P(B)} \), for \( P(B)\gt 0 \). Rearranged, it gives the multiplication rule \( P(A\cap B)=P(B)\cdot P(A\mid B) \). Events are independent when \( P(A\mid B)=P(A) \), that is, when \( P(A\cap B)=P(A)P(B) \).

Example 6: two machines

Machine X makes 60% of a factory’s items and 2% of them are defective. Machine Y makes the other 40% and 5% of them are defective. Multiply along the branches of the tree:

Adding the two defective branches: \( P(D)=0.012+0.020=0.032 \). Now suppose an item is found defective. The chance it came from Y is \( P(Y\mid D)=\dfrac{0.020}{0.032}=0.625 \). So 62.5% of defective items come from Y, even though Y makes fewer items.

Careful

\( P(A\mid B) \) and \( P(B\mid A) \) are usually different. The chance that a defective item came from Y (0.625) is not the chance that a Y item is defective (0.05).

5. The binomial distribution

A random variable \( X \) is binomial when it counts successes in \( n \) trials that are independent, have only two outcomes, and share the same success probability \( p \).

Binomial formulas

\( P(X=k)=C(n,k)\,p^{k}(1-p)^{\,n-k} \) for \( k=0,1,\dots,n \). The mean is \( \mu=np \) and the standard deviation is \( \sigma=\sqrt{np(1-p)} \).

Example 7: free throws

A player makes 40% of her shots and takes 5 independent shots. Then \( P(X=2)=C(5,2)(0.4)^2(0.6)^3=10\cdot0.16\cdot0.216=0.3456 \). The chance of at least one success is \( 1-P(X=0)=1-0.6^5=0.92224 \). The mean is \( 5\cdot0.4=2 \) shots and \( \sigma=\sqrt{5\cdot0.4\cdot0.6}=\sqrt{1.2}\approx1.10 \).

6. The normal distribution

Many measurements (heights, masses, test scores, manufacturing errors) pile up in a symmetric, bell-shaped curve centered at the mean \( \mu \). The spread is set by the standard deviation \( \sigma \), and the total area under the curve is 1.

Empirical rule (68-95-99.7)

In a normal distribution, about 68% of the values lie within \( 1\sigma \) of the mean, about 95% within \( 2\sigma \), and about 99.7% within \( 3\sigma \).

Example 8: apples

Apple masses are normal with \( \mu=150 \) g (about 5.3 oz) and \( \sigma=10 \) g. About 68% weigh between 140 g and 160 g, and about 95% between 130 g and 170 g. The fraction above 170 g is \( \dfrac{100\%-95\%}{2}=2.5\% \), because the curve is symmetric.

7. Standard deviation and z-scores

The standard deviation measures the typical distance between the data and their mean. For a whole population of \( n \) values, \( \sigma=\sqrt{\dfrac{\sum (x-\mu)^2}{n}} \). For a sample, divide by \( n-1 \) instead and write \( s \).

Example 9: computing a standard deviation

Data: 3, 5, 5, 7, 10, with mean \( \mu=6 \).

\( x \)	\( x-\mu \)	\( (x-\mu)^2 \)
3	-3	9
5	-1	1
5	-1	1
7	1	1
10	4	16

The sum of squares is 28. Population: \( \sigma=\sqrt{28/5}=\sqrt{5.6}\approx2.37 \). Sample: \( s=\sqrt{28/4}=\sqrt7\approx2.65 \).

z-score

\( z=\dfrac{x-\mu}{\sigma} \) tells how many standard deviations a value lies above (\( z\gt0 \)) or below (\( z\lt0 \)) the mean. It lets you compare values from different scales.

Example 10: who did better?

Maya scored 84 on a test with \( \mu=72 \) and \( \sigma=8 \): \( z=\dfrac{84-72}{8}=1.5 \). Leo scored 93 on a test with \( \mu=80 \) and \( \sigma=10 \): \( z=1.3 \). Maya did better relative to her group.

8. Regression and correlation

When two quantities seem related, a scatter plot shows the pattern. The least-squares line \( \hat y=mx+b \) is the line that makes the sum of squared vertical gaps as small as possible. With means \( \bar x \), \( \bar y \):

\[ m=\dfrac{\sum (x-\bar x)(y-\bar y)}{\sum (x-\bar x)^2},\qquad b=\bar y-m\bar x. \]

The correlation coefficient \( r \), always between \( -1 \) and \( 1 \), measures how close the points are to a line: \( r \) near 1 is a strong positive trend, near \( -1 \) a strong negative trend, near 0 no linear trend.

Example 11: training

After weeks of training \( x=1,2,3,4,5 \), an athlete does \( y=3,5,4,7,9 \) push-ups. Here \( \bar x=3 \), \( \bar y=5.6 \), \( \sum(x-\bar x)(y-\bar y)=14 \), \( \sum(x-\bar x)^2=10 \), and \( \sum(y-\bar y)^2=23.2 \). So \( m=1.4 \), \( b=5.6-1.4\cdot3=1.4 \), giving \( \hat y=1.4x+1.4 \), and \( r=\dfrac{14}{\sqrt{10\cdot23.2}}\approx0.92 \). Each extra week adds about 1.4 push-ups. At week 6 the model predicts \( 9.8 \).

Careful

A strong correlation does not prove that one variable causes the other. Also avoid extrapolation: predicting far outside the range of your data is unreliable.

Key takeaways

A limit describes where \( f(x) \) is heading; it ignores the value at the point. A two-sided limit exists only if both one-sided limits agree.
For rational functions at infinity, compare the degrees of numerator and denominator.
Order matters: \( P(n,r)=\dfrac{n!}{(n-r)!} \). Order does not matter: \( C(n,r)=\dfrac{n!}{r!(n-r)!} \).
\( P(A\mid B)=\dfrac{P(A\cap B)}{P(B)} \); a tree diagram multiplies along branches and adds across branches.
Binomial: \( P(X=k)=C(n,k)p^k(1-p)^{n-k} \), \( \mu=np \), \( \sigma=\sqrt{np(1-p)} \).
Normal data: about 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations; \( z=\dfrac{x-\mu}{\sigma} \).
The least-squares line has \( m=\dfrac{\sum(x-\bar x)(y-\bar y)}{\sum(x-\bar x)^2} \); \( r \) near \( \pm1 \) means a strong linear trend, but never proves causation.