📚 Probabilities using a Calculator | 使用计算器计算概率
Graphical calculators are indispensable tools in IB Mathematics, particularly for quickly computing probabilities from various distributions. The syllabus requires familiarity with binomial, normal, Poisson, geometric, negative binomial and continuous uniform distributions, among others. Using built-in probability functions not only saves time but also reduces arithmetic errors. This article explains the key calculator commands, typical workflows and common pitfalls so you can confidently tackle any probability question in Paper 2.
图形计算器在IB数学中是不可或缺的工具，尤其是在快速计算各种分布的概率时。大纲要求熟悉二项分布、正态分布、泊松分布、几何分布、负二项分布和连续均匀分布等。使用内置概率函数不仅能节省时间，还能减少计算错误。本文将解释关键的计算器命令、典型操作流程和常见陷阱，让你能够自信地应对Paper 2中的任何概率问题。
1. Binomial Distribution – Pdf and Cdf | 二项分布 – 概率密度与累积分布
s, each with success probability p. The probability mass function P(X = x) is accessed via binomPdf(n, p, x) on TI calculators (or Bpd on Casio). For cumulative probabilities such as P(X ≤ x), use binomCdf(n, p, lower, upper) x). When calculating P(X < x), remember that for discrete distributions P(X < x) = P(X ≤ x – 1). Similarly, P(X ≥ x) = 1 – P(X ≤ x – 1). Always set the correct boundaries; for example, P(3 ≤ X ≤ 7) = binomCdf(n, p, 3, 7).
二项分布描述在n次独立试验中成功次数的分布，每次成功概率为p。概率质量函数P(X = x)可通过TI计算器上的binomPdf(n, p, x)（或在Casio上使用Bpd）计算。对于累积概率例如P(X ≤ x)，使用binomCdf(n, p, lower, upper)。如果你的计算器只需要上界（如TI-84），将lower设为0即可求得P(X ≤ x)。计算P(X < x)时，记住对于离散分布有P(X < x) = P(X ≤ x – 1)。类似地，P(X ≥ x) = 1 – P(X ≤ x – 1)。务必设定正确的边界；例如，P(3 ≤ X ≤ 7) = binomCdf(n, p, 3, 7)。
On the TI-Nspire, navigate to Menu > Probability > Distributions > Binomial Pdf or Binomial Cdf. Enter the number of trials, success probability and the specific x‑value (or bounds). The result is displayed immediately. Casio users will find these functions under STAT > DIST > BINM 0.4. Here you would call binomPdf(12, 0.4, 5). To find the probability of at most 5 heads, use binomCdf(12, 0.4, 0, 5). Always double‑check whether the question says “less than” or “no more than”, as this affects the boundary value.
在TI-Nspire上，导航至菜单 > 概率 > 分布 > 二项Pdf或二项Cdf。输入试验次数、成功概率和具体的x值（或边界）。结果会立即显示。Casio用户可在STAT > DIST > BINM下找到这些功能。典型的考题可能会问：在12次抛掷一枚有偏硬币中，正面朝上的概率p=0.4，恰好出现5次正面的概率是多少。此时应调用binomPdf(12, 0.4, 5)。要计算至多5次正面的概率，则使用binomCdf(12, 0.4, 0, 5)。务必仔细审题，看清是“少于”还是“不多于”，因为这会影响边界值。
2. Normal Distribution – Calculating Probabilities | 正态分布 – 计算概率
For a continuous random variable X ~ N(μ, σ²), probabilities are found with the cumulative normal function, usually labelled normalCdf(lower, upper, μ, σ) uous, P(X = x) = 0, so we only work with intervals. To find P(X ≤ a), use a lower bound of –1099 (or a very large negative number such as –1E99) and an upper bound of a. Similarly, P(X ≥ b) uses lower = b, upper = 1099. Many calculators let you omit the tail bound by typing a very small or large number, but it is safer to explicitly enter –1099 and 1099.
对于连续型随机变量X ~ N(μ, σ²)，概率通过累积正态函数计算，通常标记为normalCdf(lower, upper, μ, σ)。由于正态分布是连续的，P(X = x) = 0，所以我们只处理区间概率。要求P(X ≤ a)，可使用下界–1099（或一个非常大的负数如–1E99），上界为a。类似地，P(X ≥ b)使用下界=b，上界=1099。许多计算器允许你直接留空尾部，但最好明确输入–1099和1099以确保准确。
On the TI‑84, press 2nd VARS (DISTR) and select 2:normalcdf(. Enter lower, upper, μ, σ in order, separated by commas. On the TI‑Nspire, go to Menu > Probability > Distributions > Normal Cdf. Casio calculators require you to navigate to STAT > DIST > NORM > Ncd. For example, if IQ scores are normally distributed with mean 100 and standard deviation 15, the proportion of people with IQs between 85 and 130 is normalCdf(85, 130, 100, 15). If the question asks for the probability that a randomly selected person has an IQ above 140, compute normalCdf(140, 1099, 100, 15). Always check whether the standard deviation or variance is given; input the standard deviation σ, not the variance σ².
在TI‑84上，按2nd VARS（DISTR）并选择2:normalcdf(。依次输入下界、上界、μ、σ，用逗号分隔。在TI‑Nspire上，前往菜单 > 概率 > 分布 > 正态Cdf。Casio计算器则需导航至STAT > DIST > NORM > Ncd。例如，若智商分数服从均值为100、标准差为15的正态分布，那么智商在85到130之间的人群比例就是normalCdf(85, 130, 100, 15)。如果题目问随机选一人智商高于140的概率，则计算normalCdf(140, 1099, 100, 15)。务必确认给出的是标准差还是方差；输入时要用标准差σ，而非方差σ²。
3. Inverse Normal – Finding x Given a Probability | 逆正态 – 给定概率求x值
The inverse normal function invNorm(area, μ, σ) returns the value x such that P(X ≤ x) = area (the left‑ bility). This is invaluable for determining quantiles, confidence interval margins and cut‑off values. Some calculators, such as the TI‑Nspire, also offer an option to find the right‑ail inverse or a symmetric central region. To find the value that corresponds to the top 5%, compute invNorm(0.95, μ, σ), because the left‑ eck whether the question requires a left or right tail. Casio’s inverse normal is labelled InvN, with the same parameters: area (left tail), μ, σ.
逆正态函数invNorm(area, μ, σ)返回满足P(X ≤ x) = area（左侧尾部的概率）的x值。这在确定分位数、置信区间边界和截断值时非常有用。某些计算器，如TI‑Nspire，还提供右尾逆求或对称中心区域的选项。要找出对应最高5%的值，应计算invNorm(0.95, μ, σ)，因为左尾面积为0.95。如果需要第99百分位数，面积就是0.99。一定要画个草图，并确认题目要求的是左尾还是右尾。Casio的逆正态标注为InvN，参数相同：面积（左尾）、μ、σ。
Often in IB questions you are given a probability like “the fastest 10% of athletes finish before time T”. Here T = invNorm(0.10, μ, σ). Notice that “fastest 10%” corresponds to the lower tail, so the left‑tail area is 0.10. If the wording is “the slowest 20%”, you want the right‑tail 20%, so the left‑ en when students carelessly choose the wrong area; always translate the phrase into a left‑tail area before typing. For a central 95% rval, the lower bound is invNorm(0.025, μ, σ) and the upper bound is invNorm(0.975, μ, σ).
在IB题目中常见这样的描述：“最快的10%运动员在时间T之前完成”。这里T = invNorm(0.10, μ, σ)。注意“最快的10%”对应左尾，所以左尾面积为0.10。如果表述为“最慢的20%”，你需要的右尾20%，所以左尾面积为0.80，T = invNorm(0.80, μ, σ)。学生常因粗心选错面积而出错；在输入之前，务必将描述转化为左尾面积。对于中心95%区间，下界为invNorm(0.025, μ, σ)，上界为invNorm(0.975, μ, σ)。
4. Poisson Distribution – Modelling Rare Events | 泊松分布 – 建模稀有事件
nterval, where λ is the mean rate. The probability mass function P(X = x) is obtained with poissonPdf(λ, x) (or Ppd on Casio). The cumulative function P(X ≤ x) is poissonCdf(λ, lower, upper) 4, 8). If your calculator only supports an upper bound for Cdf, you can still obtain lower‑ son distribution is often used for calls per hour, defects per metre or accidents per day.
泊松分布Po(λ)描述在固定区间内事件发生的次数，其中λ是平均发生率。概率质量函数P(X = x)可通过poissonPdf(λ, x)（Casio上为Ppd）获得。累积函数P(X ≤ x)为poissonCdf(λ, lower, upper)。由于泊松分布是离散的，同样的边界技巧适用：P(X < 5) = poissonCdf(λ, 0, 4)，而P(3 < X ≤ 8) = poissonCdf(λ, 4, 8)。如果你的计算器Cdf只支持上界，仍可通过设lower = 0得到左尾概率。泊松分布常用于每小时电话数、每米缺陷数或每日事故数等场景。
On the TI‑Nspire, Poisson functions are under Menu > Probability > Distributions > Poisson Pdf / Poisson Cdf r hour, the probability of receiving exactly 4 emails in the next hour is poissonPdf(2.5, 4). The probability of receiving at most 3 emails is poissonCdf(2.5, 0, 3). Casio users will find the commands in STAT > DIST > POISN e question asks about half a day, adjust λ accordingly (e.g. new λ = 1.25 for half a day).
在TI‑Nspire上，泊松函数位于菜单 > 概率 > 分布 > 泊松Pdf / 泊松Cdf下。输入λ和具体的值或边界。例如，如果λ = 2.5封邮件/小时，下一小时内恰好收到4封邮件的概率为poissonPdf(2.5, 4)。至多收到3封邮件的概率为poissonCdf(2.5, 0, 3)。Casio用户可在STAT > DIST > POISN中找到命令。当λ不是整数时，计算方式完全相同。考试中常见的延伸是利用比例性质：如果给出每天的率，但问题要求的是半天的情况，则相应调整λ（例如半天的λ = 1.25）。
5. Geometric Distribution – First Success | 几何分布 – 首次成功
e geometric distribution models the number of trials needed to obtain the first success (X ~ ntions exist: one where X is the number of trials including the success (support x = 1, 2, … d another where X is the number of failures before the first success (support x = 0, 1, 2, …). Make sure you know which version your syllabus uses. For the “trials to first success” model, the pdf functions are geometPdf(p, x) and geometCdf(p, lower, upper). To find P(X ≤ k), enter geometCdf(p, 1, k). For P(X > k) = (1 – p)k, you can either use the formula or compute 1 – geometCdf(p, 1, k). The geometric distribution is memoryless, a property sometimes tested.
几何分布模拟获得首次成功所需的试验次数（X ~ Geo(p)）。在IB中，有两种约定：一种是将X定义为包括成功在内的试验次数（取值x = 1, 2, …），另一种是将X定义为首次成功之前的失败次数（取值x = 0, 1, 2, …）。务必确认你的大纲使用的是哪个版本。对于“首次成功的试验次数”模型，概率函数为geometPdf(p, x)和geometCdf(p, lower, upper)。要计算P(X ≤ k)，输入geometCdf(p, 1, k)。对于P(X > k) = (1 – p)k，你可以直接使用公式，或计算1 – geometCdf(p, 1, k)。几何分布具有无记忆性，这点有时也会考查。
t bulb passes a quality test is 0.9. What is the probability that the first failure occurs on the 5th test? nd you would compute geometPdf(0.1, 5). To find the probability that the first failure occurs by the 3rd test, compute geometCdf(0.1, 1, 3). If the order were reversed (p as success), be careful because the “first success” language swaps roles. Always label p as the probability of the event you count as “success” -Nspire, commands are in the same probability menu. Casio users may need to install an add‑in or use the distribution menu that includes GeoPD and GeoCD.
举例：一个灯泡通过质检的概率是0.9。首次不合格恰好在第5次测试出现的概率是多少？这里p = 0.1（不合格概率），你应该计算geometPdf(0.1, 5)。要计算首次不合格出现在前3次测试之内的概率，使用geometCdf(0.1, 1, 3)。如果角色颠倒（p定义为成功概率），要格外小心，因为“首次成功”的表述会互换角色。始终将p标记为你所等待的那个事件（你所计数的“成功”）的概率。在TI-Nspire上，命令位于相同的概率菜单中。Casio用户可能需要安装附加软件，或使用包含GeoPD和GeoCD的分布菜单。
6. Negative Binomial Distribution – k‑th Success | 负二项分布 – 第k次成功
The negative binomial distribution X ~ NB(k, p) (or r instead of k) counts the number of trials needed to achieve the k‑th success. In IB, the support usually starts at x = k, k+1, k+2, … . The probability of exactly x trials is given by nbinPdf(k, p, x), and the cumulative probability is nbinCdf(k, p, lower, upper) , some calculators define the distribution in terms of the number of failures before the k‑ your calculator’s manual carefully. As with the geometric, the IB syllabus may specify the “trials to k‑th success” tation. To compute P(X ≤ m), use nbinCdf(k, p, k, m). For P(X < m), use nbinCdf(k, p, k, m‑1).
负二项分布X ~ NB(k, p)（有时用r代替k）计数达到第k次成功所需的试验次数。在IB中，定义域通常从x = k, k+1, k+2, …开始。恰好需要x次试验的概率由nbinPdf(k, p, x)给出，累积概率为nbinCdf(k, p, lower, upper)。不过，有些计算器将分布定义为第k次成功之前的失败次数；请仔细阅读计算器的说明书。与几何分布一样，IB大纲可能指定“达到第k次成功所需的试验次数”这一解释。要计算P(X ≤ m)，使用nbinCdf(k, p, k, m)。对于P(X < m)，使用nbinCdf(k, p, k, m‑1)。
For example, a basketball player has a free‑ df(5, 0.7, 8). The probability she requires at most 10 attempts to reach the 5th success is nbinCdf(5, 0.7, 5, 10). These functions appear under the same probability menu in TI-Nspire. When using a TI-84, you might need to enter the number of successes, the probability, and then the x-value. Some older calculators do not have a built‑ t case, you can fall back on the relationship to the binomial distribution: P(X = x) = C(x‑1, k‑1) * pk * (1‑p)x‑k. You can compute this manually using nCr if needed.
例如，一名篮球运动员罚球命中率为0.7。试求她恰好需要8次尝试才能投中第5个球所需的概率。此处k=5, p=0.7, x=8，所以使用nbinPdf(5, 0.7, 8)。她至多需要10次尝试就能实现第5次成功的概率为nbinCdf(5, 0.7, 5, 10)。这些函数在TI-Nspire的概率菜单中均可找到。使用TI-84时，你可能需要依次输入成功次数、概率和x值。有些较旧的计算器没有内置负二项函数；此时可借助与二项分布的关系：P(X = x) = C(x‑1, k‑1) × pk × (1‑p)x‑k。必要时可使用nCr手动计算。
7. Continuous Uniform Distribution – Equal Likelihood Intervals | 连续均匀分布 – 等可能区间
A continuous uniform distribution X ~ bility density over [a, b]. While dedicated uniform functions are not always present on IB‑ trick or simple arithmetic. The cumulative probability P(X ≤ x) for any x in [a, b] is (x – a)/(b – a). To find P(c ≤ X ≤ d), you can use the uniform Cdf if available, or simply type (d – c)/(b – a) on the home screen. Some TI calculators have a uniformCdf(lower, upper, a, b) command accessible via the catalog or an additional app. On the TI‑Nspire, uniform distribution functions are not built into the standard probability menu, but you can define your own function or just use formula entry. Casio’s Distribution mode does not include a uniform Cdf, but the operation is trivial with basic arithmetic.
连续均匀分布X ~ U(a, b)在区间[a, b]上具有恒定的概率密度。尽管经IB认证的计算器未必都配有专门的均匀分布函数，但使用正态Cdf技巧或简单算术即可轻松计算概率。对于[a, b]中的任意x，累积概率P(X ≤ x) = (x – a)/(b – a)。要计算P(c ≤ X ≤ d)，如果可用的话可以使用均匀Cdf，或者直接在主屏幕输入(d – c)/(b – a)。部分TI计算器通过目录或附加应用提供uniformCdf(lower, upper, a, b)命令。TI‑Nspire的标准概率菜单没有内置均匀分布函数，但你可以自行定义函数或直接使用公式输入。Casio的分布模式不包含均匀Cdf，但用基本算术操作也是小事一桩。
The inverse uniform problem is also straightforward: given a probability p (left‑tail area), the corresponding value x is x = a + p(b – waiting time for a bus is uniformly distributed between 2 and 10 minutes, the time that 30% of passengers wait longer than is the 70th percentile: x = 2 + 0.7(10 – the correct order, and remember that P(X = exact value) = 0 for continuous distributions.
逆均匀问题同样简单：给定概率p（左尾面积），对应的值x为x = a + p(b – a)。例如，如果公交车的等待时间在2到10分钟之间服从均匀分布，那么30%的乘客等待超过的时间（即第70百分位数）为：x = 2 + 0.7(10 – 2) = 7.6分钟。在考试中，你可以在写出公式后直接写出计算式子，无需特殊按键。不过，始终要确保a和b的顺序正确，并记住对于连续分布，P(X = 精确值) = 0。
8. Key Tips and Common Pitfalls | 关键提示与常见陷阱
Always distinguish between Pdf (gives the exact probability for a discrete variable or density height for a continuous one) and Cdf distributions, Pdf is rarely used; stick to Cdf for intervals. When using invNorm, double‑check that you are entering the left‑ tally leave it in radian or degree mode but that does not affect probability calculations directly, though in statistics the mode for angles is irrelevant. However, check that your calculator is not set to “Stats Wizard” off if you need the interactive prompts.
务必区分Pdf（给出离散变量的精确概率或连续变量的密度高度）和Cdf（给出累积概率）。对离散分布，Pdf用于计算P(X = x)，而Cdf处理P(X ≤ x)。对于连续分布，极少使用Pdf；计算区间概率时坚持使用Cdf。使用invNorm时，仔细核对输入的是左尾面积。在草稿纸上快速画个图，直观显示所需区域。确认计算器处于正确的模式；有些学生无意中将角度单位留在弧度或度模式，尽管这对概率计算没有直接影响——统计中角度模式无关紧要。但如果你需要交互式提示，请检查计算器未关闭“统计向导”。
Another common error is misinterpreting the phrase “fewer than” as ≤ instead of <. For example, “fewer than 5” means x = 0,1,2,3,4. Therefore, on a binomial Cdf you must use upper = 4, not 5. “At least 7” means x ≥ 7, so you must use lower = 7 or compute 1 – Cdf(… nσ²). Use your calculator’s memory and list features to store results, especially in multi‑ f with the calculator’s syntax for infinity; many models accept 1E99 as a proxy for +∞ and –1E99 for –∞.
另一个常见错误是将“少于5”误解为≤而不是<。例如，“fewer than 5”意味着x = 0,1,2,3,4。因此，在二项Cdf中你必须使用上界 = 4，而非5。“至少7”表示x ≥ 7，所以必须使用下界=7或计算1 – Cdf(…, 6)。在泊松和几何分布中同样适用此逻辑。当题目给出样本均值并询问总和的分布时，记得调整参数（例如，独立同分布正态变量之和的均值为nμ，方差为nσ²）。利用计算器的存储和列表功能保存结果，尤其在多步问题中。最后，熟悉计算器表示无穷的语法；许多型号接受1E99作为+∞的替代，–1E99作为–∞的替代。
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