How is $P(a \le X \le b)$ calculated using CDF values?

It is calculated as $P(X \le b) - P(X \le a - 1)$. This takes the cumulative probability up to the upper bound and subtracts the cumulative probability of everything below the lower bound.

How do you calculate $P(X \ge k)$ for a binomial distribution using cumulative tables?

Since tables provide $P(X \le x)$, you must use the complement identity $P(X \ge k) = 1 - P(X \le k - 1)$. This excludes all integer values strictly less than $k$ from the total probability of $1$.

What is the difference between $P(X < 5)$ and $P(X \le 5)$ in a binomial context?

$P(X \le 5)$ includes the probability of exactly $5$ successes, whereas $P(X < 5)$ does not. Because the distribution is discrete, $P(X < 5)$ is equivalent to $P(X \le 4)$.

When should you model failures ($Y$) instead of successes ($X$)?

This is typically done when the probability of success $p$ is greater than $0.5$, as standard binomial tables only list values for $p \le 0.5$. By modeling failures with $1-p$, the table becomes usable.

What common mistake occurs when calculating $P(X > k)$?

Students often use $1 - P(X \le k-1)$, which incorrectly includes the value $k$. The correct identity is $1 - P(X \le k)$, as $P(X > k)$ requires all values strictly greater than $k$.

Why is it an error to ignore the binomial coefficient $\binom{n}{r}$ in the PMF formula?

The coefficient accounts for all possible permutations of successes and failures across $n$ trials. Without it, you are only calculating the probability of one specific sequence (e.g., all successes first, then all failures).

What happens if you use continuous distribution logic (like a normal distribution) for binomial bounds?

You will likely miss the 'discrete jump' between integers. For binomials, $P(X \le 4.5)$ is simply $P(X \le 4)$; treating it as a continuous range without adjustment leads to precision errors.

What is the formula for the probability of exactly $r$ successes?

The formula is $P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$. It combines the number of combinations, the probability of successes, and the probability of failures.

Define the term 'Cumulative Distribution Function' (CDF) for a binomial variable.

The CDF is the probability that the random variable $X$ takes a value less than or equal to $x$. Mathematically, it is the sum $P(X \le x) = \sum_{i=0}^{x} P(X = i)$.

Why is $P(X = 2.5)$ always zero in a binomial distribution?

The binomial distribution is a discrete distribution that counts 'trials'. You cannot have a non-integer number of successes (like half a success) in a fixed number of whole trials.

Library Podcasts

Courses

Referral & Rewards

Revision Notes

International A-Level

Pearson Edexcel

Maths

Statistics 2

Statistical Distributions

Calculating Binomial Probabilities

Summary

Calculating binomial probabilities involves determining the likelihood of a specific number of successes in a fixed number of independent trials. This is achieved using the Probability Mass Function (PMF) for individual values or the Cumulative Distribution Function (CDF) for ranges, often simplified through specific logical identities and statistical tables.

1. The Probability Mass Function (PMF)

2. Cumulative Probabilities and Tables

The Cumulative Distribution Function (CDF), denoted $P(X \le x)$ , represents the sum of probabilities for all outcomes from $0$ up to and including $x$ . This is essential for finding the probability of 'at most' a certain number of successes.
Statistical Tables provide pre-calculated CDF values for standard parameters $n$ and $p$ . These tables are typically limited to $p \le 0.5$ to maintain conciseness, requiring symmetry rules for higher probabilities.
When using tables, results are usually rounded to four decimal places. If a specific $p$ or $n$ value is not in the table, the PMF formula must be used manually or through a specialized calculator.

A diagram illustrating the range probability identity P(a ≤ X ≤ b) = P(X ≤ b) - P(X ≤ a-1) on a discrete number line.

3. Handling Probabilities Greater Than 0.5

4. Logical Identities for Inequality Ranges

5. Key Distinctions

6. Exam Strategy & Tips