IB Mathematics: Using Calculators to Compute Probabilities | IB数学：利用计算器求概率的方法

📚 IB Mathematics: Using Calculators to Compute Probabilities | IB数学：利用计算器求概率的方法

In IB Mathematics, a Graphic Display Calculator (GDC) is not just allowed — it is an essential tool. Probability questions on Paper 2 often involve calculations that are tedious or impossible by hand, such as binomial sums, normal tail probabilities, or inverse-normal quantiles. Mastering the calculator’s probability functions saves time, reduces arithmetic errors, and unlocks full marks on application-style problems.

在IB数学中，图形计算器（GDC）不仅是允许使用的工具，更是不可或缺的工具。Paper 2中的概率题往往涉及手算极其繁琐甚至不可能完成的计算，例如二项分布累加、正态分布尾概率或逆正态分位数。熟练掌握计算器的概率功能，既能节省时间、减少运算错误，也能在应用题中轻松拿到满分。

1. Calculator Fundamentals | 计算器基础

Before solving any probability problem, you must know where the relevant commands are located. On the TI-Nspire CX, open the Calculator application, press menu → 5: Probability → 5: Distributions. On Casio models (e.g. fx-9750GII), press OPTN → F5: DIST. The menu names may vary, but the underlying functions — binomial, Poisson, normal, inverse normal — are universal.

在解决任何概率问题之前，你必须知道相关命令所在的位置。在TI-Nspire CX上，打开计算器应用，依次按menu → 5: Probability → 5: Distributions。在卡西欧机型（如fx-9750GII）上，按OPTN → F5: DIST。菜单名称可能略有差异，但底层的功能——二项分布、泊松分布、正态分布、逆正态——是通用的。

Always set the calculator to the correct mode: Radians or Degrees are irrelevant for probability, but you should ensure that the probability functions are available. For the TI-Nspire, you must also know whether you are using the “Pdf” or “Cdf” version. A “Pdf” gives P(X = x) for discrete distributions, while a “Cdf” gives P(X ≤ x) or P(a ≤ X ≤ b).

始终将计算器设置为正确的模式：弧度或角度对于概率并不重要，但你需要确保概率函数可用。对于TI-Nspire，你还需要分清使用的是“Pdf”还是“Cdf”版本。“Pdf”用于离散分布中求P(X = x)，而“Cdf”用于求P(X ≤ x)或P(a ≤ X ≤ b)。

2. Permutations, Combinations and Factorials | 排列、组合与阶乘

Many probability questions use counting methods. The binomial coefficient nCr and the permutation function nPr are built into every GDC. On the TI-Nspire, type nCr(10, 3) or use menu → 5: Probability → 3: Combinations. On Casio, press OPTN → F6 → F3: PROB to find nCr and nPr.

许多概率问题使用计数方法。二项系数nCr和排列函数nPr内置于每一台GDC中。在TI-Nspire上，输入nCr(10, 3)，或使用menu → 5: Probability → 3: Combinations。在卡西欧机型上，按OPTN → F6 → F3: PROB可找到nCr和nPr。

Function	Meaning	Example
nPr(n, r)	Ordered arrangements	nPr(5, 2) = 20
nCr(n, r)	Unordered selections	nCr(5, 2) = 10
n!	Factorial	5! = 120

Always verify whether the problem asks for ordered arrangements (permutations) or unordered groups (combinations). For example, choosing a committee of 3 from 10 uses C(10,3), while arranging 3 books on a shelf from 10 uses P(10,3).

始终要确认题目要求的是有序排列（排列数）还是无序组合（组合数）。例如，从10人中选3人组成委员会使用C(10,3)，而从10本书中选3本放在书架上排列则使用P(10,3)。

3. Probability Distributions on Your GDC | 计算器上的概率分布

Your GDC can handle four main distributions: binomial, Poisson, normal, and (on most models) Student’s t. For IB Mathematics SL/HL, the first three are essential. Each distribution appears in the distributions menu with two commands: Pdf and Cdf.

你的计算器可以处理四种主要分布：二项分布、泊松分布、正态分布，以及（在大多数机型上）学生t分布。对于IB数学SL/HL，前三者是必需的。每种分布都在分布菜单中对应两个命令：Pdf和Cdf。

For discrete distributions, “Pdf” computes the probability at a single value, e.g. P(X = 4). “Cdf” computes cumulative probabilities, e.g. P(X ≤ 4) or P(2 ≤ X ≤ 6). For continuous distributions like the normal, the “Pdf” gives the height of the curve (rarely used), while the “Cdf” gives the area under the curve — the actual probability.

对于离散分布，“Pdf”计算单个取值的概率，例如P(X = 4)。“Cdf”计算累积概率，例如P(X ≤ 4)或P(2 ≤ X ≤ 6)。对于正态分布这样的连续分布，“Pdf”给出曲线的高度（很少使用），而“Cdf”给出曲线下的面积——即真实的概率。

4. Binomial Distribution | 二项分布

The binomial distribution models the number of successes in n independent Bernoulli trials, each with success probability p. The probability function is:

二项分布用于描述n次独立伯努利试验中成功次数的模型，每次成功概率为p。其概率函数为：

P(X = x) = C(n, x) · pˣ · (1 − p)ⁿ⁻ˣ

On a TI-Nspire, use menu → 5: Probability → 5: Distributions → BinomPdf (for exact value) or BinomCdf (for cumulative). On a Casio, go to OPTN → F5: DIST → F5: BINM. Enter n, p, and the value x.

在TI-Nspire上，使用menu → 5: Probability → 5: Distributions → BinomPdf（精确值）或BinomCdf（累积值）。在卡西欧机型上，进入OPTN → F5: DIST → F5: BINM，输入n、p和x值。

Key IB questions often ask for P(X ≤ k), P(X ≥ k), or P(a ≤ X ≤ b). Use Cdf for these. Remember that P(X ≥ k) = 1 − P(X ≤ k−1), which you can compute directly with BinomCdf or by using the complement rule.

IB常见题型会问P(X ≤ k)、P(X ≥ k)或P(a ≤ X ≤ b)。这些应使用Cdf。注意P(X ≥ k) = 1 − P(X ≤ k−1)，你可以直接用BinomCdf计算，也可以用互补事件法则处理。

Example: Let X ~ B(12, 0.4). Find P(3 ≤ X ≤ 6). On a TI-Nspire, use BinomCdf(12, 0.4, 3, 6) → 0.7425 (to 4 s.f.). On Casio, select Binomial C.D. with lower=3, upper=6.

例如：设X ~ B(12, 0.4)，求P(3 ≤ X ≤ 6)。在TI-Nspire上，使用BinomCdf(12, 0.4, 3, 6)，结果为0.7425（保留4位有效数字）。在卡西欧上，选择Binomial C.D.并设置lower=3、upper=6。

5. Poisson Distribution | 泊松分布

The Poisson distribution models the number of events occurring in a fixed interval of time or space, given the average rate λ. The probability function is:

泊松分布用于描述在固定时间或空间间隔内事件发生的次数，给定平均发生率λ。其概率函数为：

P(X = x) = e⁻λ · λˣ / x!

On a GDC, look for PoissonPdf and PoissonCdf. For example, if X ~ Po(5), then P(X = 3) is found by PoissonPdf(5, 3) ≈ 0.1404. The cumulative probability P(X ≤ 2) is PoissonCdf(5, 2) ≈ 0.1247.

在GDC上，查找PoissonPdf和PoissonCdf。例如，若X ~ Po(5)，则P(X = 3)可通过PoissonPdf(5, 3)求得，约为0.1404。累积概率P(X ≤ 2)为PoissonCdf(5, 2)，约为0.1247。

Be careful with the parameter: λ is the mean and also the variance. In IB questions, λ is often given per unit time. If the interval changes, adjust λ accordingly. For example, if λ = 6 per hour, then for 30 minutes λ = 3.

注意参数：λ既是均值也是方差。在IB题目中，λ通常按单位时间给出。如果时间段改变，需要相应调整λ。例如，若λ = 6次/小时，那么30分钟内λ = 3。

6. Normal Distribution | 正态分布

The normal distribution is a continuous distribution defined by its mean μ and standard deviation σ. On a GDC, use the Normal Cdf command to find probabilities. On TI-Nspire: menu → 5: Probability → 5: Distributions → Normal Cdf. Enter the lower bound, upper bound, μ, and σ.

正态分布是一种连续分布，由均值μ和标准差σ定义。在GDC上，使用Normal Cdf命令求概率。在TI-Nspire上：menu → 5: Probability → 5: Distributions → Normal Cdf，输入下限、上限、μ和σ。

To find P(X < a), use a very small lower bound, such as −10⁹⁹ or −999999. To find P(X > b), use a very large upper bound, such as 10⁹⁹. For example, if X ~ N(20, 5²), then P(18 < X < 22) is found by entering lower=18, upper=22, μ=20, σ=5. The result is ≈ 0.3108.

要求P(X < a)，需使用一个非常小的下限，如−10⁹⁹或−999999。要求P(X > b)，则使用一个非常大的上限，如10⁹⁹。例如，若X ~ N(20, 5²)，求P(18 < X < 22)时，输入lower=18、upper=22、μ=20、σ=5，结果约为0.3108。

Always standardise if you are given a standard normal table, but with a GDC you can enter the original μ and σ directly. This avoids rounding errors when converting to z-scores.

如果你手边是标准正态表，通常需要标准化；但使用GDC时可以直接输入原始的μ和σ，这样可以避免将数据转化为z分数时产生的舍入误差。

7. Inverse Normal (Quantiles) | 逆正态（分位数）

Inverse normal problems ask for the value of x that corresponds to a given cumulative probability. This is essential in IB questions about finding the passing mark, the 95th percentile, or a critical value for a hypothesis test.

逆正态问题要求给定累积概率反求x值。这在IB题目中非常重要，例如求及格线、第95百分位数或假设检验的临界值。

On a TI-Nspire, use invNorm(p, μ, σ) where p is the cumulative area to the left of x. On a Casio, use OPTN → F5: DIST → F3: NORM → F3: InvN. Always remember that the “area” is the left-tail probability unless you specify otherwise.

在TI-Nspire上，使用invNorm(p, μ, σ)，其中p是x左侧的累积面积。在卡西欧机型上，使用OPTN → F5: DIST → F3: NORM → F3: InvN。务必记住“面积”默认是左尾概率，除非特别说明。

Example: X ~ N(100, 15²). Find the 90th percentile. Use invNorm(0.9, 100, 15) ≈ 119.2. This means P(X ≤ 119.2) = 0.9.

例如：X ~ N(100, 15²)，求第90百分位数。使用invNorm(0.9, 100, 15)，结果约为119.2。这意味着P(X ≤ 119.2) = 0.9。

For two-sided symmetric probabilities (e.g. middle 80%), find the left tail (1−0.8)/2 = 0.1 and right tail 0.9, then use invNorm for both.

对于对称的双尾概率（如中间80%），分别计算左尾(1−0.8)/2 = 0.1和右尾0.9，然后对两者使用逆正态函数。

8. Using the Graphical Display | 图形显示与区间概率

Most GDCs let you visualise a probability as the area under a distribution curve. In the TI-Nspire, you can use the Graphs application, define a function like f1(x) = normPdf(x, μ, σ), and then use menu → 6: Analyze Graph → 6: Integral to shade an interval. This is useful for sanity-checking your answers and for understanding the meaning of a probability.

大多数GDC允许你将概率可视化为分布曲线下的面积。在TI-Nspire中，可以使用图形应用，定义函数如f1(x) = normPdf(x, μ, σ)，然后使用menu → 6: Analyze Graph → 6: Integral来对某个区间着色求面积。这有助于验证你的答案，并帮助你理解概率的含义。

For discrete distributions, the graphical display shows vertical bars whose heights sum to 1. You can use the table feature to list P(X = x) for all x and quickly identify the mode or find a range probability.

对于离散分布，图形显示的是垂直条形，其高度总和为1。你可以使用表格功能列出所有x对应的P(X = x)，快速识别众数或计算区间概率。

When using the graph, always check the window settings. For a normal curve with mean μ, set the x-axis from μ − 4σ to μ + 4σ so that the entire relevant area is visible.

使用图形时，务必检查窗口设置。对于均值为μ的正态曲线，将x轴设为μ − 4σ至μ + 4σ，以确保相关区域完整可见。

9. Verification and Empirical Probability | 验证与经验概率

Your GDC can also store datasets and compute empirical probabilities from frequency tables. Use the statistics mode to enter data in List1 and List2, then call menu → 4: Statistics → 1: Stat Calculations to find the mean, variance, and even the probability of an observed event as a relative frequency.

你的计算器还可以存储数据集，并依据频率表计算经验概率。在统计模式下将数据输入List1和List2，然后使用menu → 4: Statistics → 1: Stat Calculations来求均值、方差，甚至以相对频率的形式计算观测事件的概率。

To verify a theoretical probability, you can run a quick simulation. On the TI-Nspire, use randBin(n, p) or randNorm(μ, σ) to generate random values, then use countIf() to estimate the proportion of results in an interval. This will usually be close to the theoretical value if the number of trials is large.

为了验证理论概率，你可以进行快速模拟。在TI-Nspire上，使用randBin(n, p)或randNorm(μ, σ)生成随机值，然后用countIf()估计落在某个区间内的结果比例。当试验次数足够大时，该比例通常接近理论值。

In IB Paper 2, you may be asked to compare theoretical and experimental probabilities. Use the calculator’s 1-Variable Statistics to find the experimental mean, then calculate the theoretical mean from the distribution formula. A small difference suggests random error, not a flawed model.

在IB Paper 2中，你可能会被要求比较理论概率与实验概率。使用计算器的单变量统计功能求实验均值，再根据分布公式求理论均值。若两者差异较小，说明差异源于随机误差，而不是模型有误。

10. Common Pitfalls and Good Practice | 常见错误与良好实践

One common mistake is using a “Pdf” when a “Cdf” is required. For continuous distributions, a single point has zero probability, so you must always use intervals. For discrete distributions, double-check whether the question says “at least”, “more than”, “at most”, or “less than” because the inequality sign changes the bounds by 1.

一个常见错误是在需要使用“Cdf”时误用了“Pdf”。对于连续分布，单点概率为零，因此必须使用区间。对于离散分布，要仔细辨别题目说的是“至少”“多于”“至多”还是“少于”，因为不等号方向会使边界移动1个单位。

Another pitfall: entering the standard deviation instead of the variance. In IB notation, N(μ, σ²) is often written with the variance second. But in the calculator’s normal Cdf command, you must enter σ, not σ². For example, if X ~ N(20, 25), then σ = 5, and you must enter 5, not 25.

另一个陷阱是误将方差当标准差输入。在IB记号中，N(μ, σ²)的第二个参数通常是方差。但在计算器的正态Cdf命令中，必须输入σ而不是σ²。例如，若X ~ N(20, 25)，则σ = 5，应输入5而不是25。

Always write down the distribution you are using and the calculator command in your working. IB marks are often awarded for method, even if the final answer is wrong. Showing “P(X<5) = normalCdf(−9999, 5, μ, σ)" gives clear evidence of your approach.

在解题过程中，务必写下你使用的分布和计算器命令。IB评分通常看重方法，即使最终答案错误也会给过程分。写出“P(X<5) = normalCdf(−9999, 5, μ, σ)”能明确展示你的思路。

Finally, round only at the final step. Store intermediate results in the calculator’s answer memory or use exact fractions when possible. This avoids extra rounding error, especially in multi-part questions where subsequent parts use earlier results.

最后，只在最后一步进行舍入。将中间结果储存在计算器的答案存储器中，或者在可能时使用精确分数。这样能避免多余的舍入误差，尤其是在后续小题会用到前一步结果的连续题中。

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IB Mathematics: Using Calculators to Compute Probabilities | IB数学：利用计算器求概率的方法