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Statistics

Statistical Distributions: Binomial, Normal, and Approximations

Year 12 · Year 13

  • ✓By the end of this lesson students will be able to understand and apply the Binomial distribution to model discrete random variables.
  • ✓By the end of this lesson students will be able to understand and apply the Normal distribution to model continuous random variables.
  • ✓By the end of this lesson students will be able to calculate probabilities for Binomial and Normal distributions using appropriate methods, including standardisation.
  • ✓By the end of this lesson students will be able to recognise when a Binomial distribution can be approximated by a Normal distribution.
  • ✓By the end of this lesson students will be able to apply the Normal approximation to the Binomial distribution, including the use of a continuity correction.

Key concepts

Binomial Distribution

The Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant for each trial. It is a discrete probability distribution. A random variable X following a Binomial distribution is denoted as X ~ B(n, p), where 'n' is the number of trials and 'p' is the probability of success in a single trial.

P(X = r) = (nCr) * p^r * (1-p)^(n-r) Expected Value (Mean): E(X) = np Variance: Var(X) = np(1-p)
Normal Distribution

The Normal distribution is a continuous probability distribution that is symmetrical about its mean, forming a bell-shaped curve. It is characterised by two parameters: the mean (μ) and the variance (σ²). A random variable X following a Normal distribution is denoted as X ~ N(μ, σ²). The total area under the curve is equal to 1. The Standard Normal distribution, denoted as Z ~ N(0, 1), is a special case where the mean is 0 and the variance is 1. Any Normal variable X can be standardised to Z using the formula Z = (X - μ) / σ.

Z = (X - μ) / σ
Normal Approximation to the Binomial Distribution

Under certain conditions, a Binomial distribution can be approximated by a Normal distribution. This approximation is useful when 'n' is large, making direct Binomial calculations cumbersome. The conditions for a good approximation are typically np > 5 and n(1-p) > 5. When approximating, the discrete Binomial variable X is treated as a continuous Normal variable. The parameters for the approximating Normal distribution are μ = np and σ² = np(1-p). A crucial step in this approximation is the 'continuity correction', which adjusts the discrete probability for a continuous distribution.

If X ~ B(n, p) and np > 5, n(1-p) > 5, then X ≈ Y ~ N(np, np(1-p)) Continuity Correction: P(X = r) ≈ P(r - 0.5 < Y < r + 0.5) P(X ≥ r) ≈ P(Y ≥ r - 0.5) P(X > r) ≈ P(Y ≥ r + 0.5) P(X ≤ r) ≈ P(Y ≤ r + 0.5) P(X < r) ≈ P(Y ≤ r - 0.5)

Key facts to remember

  • 1The Binomial distribution X ~ B(n, p) models the number of successes in 'n' independent trials with success probability 'p'.
  • 2The probability mass function for Binomial is P(X=r) = (nCr) * p^r * (1-p)^(n-r).
  • 3The mean of a Binomial distribution is E(X) = np, and the variance is Var(X) = np(1-p).
  • 4The Normal distribution X ~ N(μ, σ²) is a continuous, symmetrical, bell-shaped distribution defined by its mean (μ) and variance (σ²).
  • 5To standardise a Normal variable X to Z, use the formula Z = (X - μ) / σ, where Z ~ N(0, 1).
  • 6A Binomial distribution B(n, p) can be approximated by a Normal distribution N(np, np(1-p)) if np > 5 and n(1-p) > 5.
  • 7When using a Normal approximation for a discrete Binomial variable, a continuity correction of ±0.5 must be applied to the boundaries.
  • 8For P(X=r) in Binomial, use P(r-0.5 < Y < r+0.5) in Normal approximation. For P(X≤r), use P(Y ≤ r+0.5).

Worked examples

Example 1

A fair coin is tossed 10 times. Let X be the number of heads obtained. Calculate the probability of getting exactly 7 heads.

IIdentify the distribution: This is a Binomial distribution as there are a fixed number of trials (n=10), two outcomes (head/tail), independent trials, and a constant probability of success (p=0.5 for a fair coin). So, X ~ B(10, 0.5).
IIIdentify the required probability: We need to find P(X = 7).
IIIApply the Binomial probability formula: P(X = r) = (nCr) * p^r * (1-p)^(n-r)
IVSubstitute the values: P(X = 7) = (10C7) * (0.5)^7 * (1-0.5)^(10-7)
VCalculate the combination: 10C7 = 10! / (7! * 3!) = (10 * 9 * 8) / (3 * 2 * 1) = 120
VICalculate the powers: (0.5)^7 = 0.0078125, (0.5)^3 = 0.125
VIIMultiply the values: P(X = 7) = 120 * 0.0078125 * 0.125 = 0.1171875

Answer

P(X = 7) = 0.117 (to 3 significant figures)

For exam questions, it is often acceptable to use a calculator's Binomial PD function directly after stating the distribution and parameters.

Example 2

The heights of adult males in a certain town are Normally distributed with a mean of 175 cm and a standard deviation of 8 cm. Find the probability that a randomly selected adult male has a height between 170 cm and 185 cm.

IIdentify the distribution parameters: X ~ N(175, 8²), so μ = 175 and σ = 8.
IIIdentify the required probability: P(170 < X < 185).
IIIStandardise the values using Z = (X - μ) / σ:
IVFor X = 170: Z1 = (170 - 175) / 8 = -5 / 8 = -0.625
VFor X = 185: Z2 = (185 - 175) / 8 = 10 / 8 = 1.25
VIRewrite the probability in terms of Z: P(-0.625 < Z < 1.25).
VIIUse Normal distribution tables or a calculator: P(-0.625 < Z < 1.25) = P(Z < 1.25) - P(Z < -0.625).
VIIIFrom tables/calculator: P(Z < 1.25) ≈ 0.89435
9From tables/calculator: P(Z < -0.625) = 1 - P(Z < 0.625) ≈ 1 - 0.73401 = 0.26599 (using interpolation for 0.625 or direct calculator value).
10Calculate the difference: 0.89435 - 0.26599 = 0.62836

Answer

The probability is 0.628 (to 3 significant figures).

Always sketch the Normal curve to visualise the area you are calculating. Be careful with negative Z-scores and using symmetry.

Example 3

A biased coin is tossed 100 times. The probability of getting a head is 0.6. Use a Normal approximation to estimate the probability of getting between 55 and 65 heads (inclusive).

IIdentify the Binomial distribution: X ~ B(100, 0.6). So n = 100, p = 0.6.
IICheck conditions for Normal approximation: np = 100 * 0.6 = 60. n(1-p) = 100 * 0.4 = 40. Both are > 5, so a Normal approximation is appropriate.
IIIDetermine parameters for the approximating Normal distribution: μ = np = 60. σ² = np(1-p) = 100 * 0.6 * 0.4 = 24. So σ = √24 ≈ 4.89898.
IVState the approximating distribution: Y ~ N(60, 24).
VApply continuity correction: We need P(55 ≤ X ≤ 65). For a continuous approximation, this becomes P(54.5 ≤ Y ≤ 65.5).
VIStandardise the values:
VIIFor Y = 54.5: Z1 = (54.5 - 60) / √24 = -5.5 / 4.89898 ≈ -1.1226
VIIIFor Y = 65.5: Z2 = (65.5 - 60) / √24 = 5.5 / 4.89898 ≈ 1.1226
9Rewrite the probability in terms of Z: P(-1.1226 < Z < 1.1226).
10Use Normal distribution tables or a calculator: P(-1.1226 < Z < 1.1226) = P(Z < 1.1226) - P(Z < -1.1226).
11P(Z < 1.1226) ≈ 0.8692 (using calculator or interpolating from tables).
12P(Z < -1.1226) = 1 - P(Z < 1.1226) ≈ 1 - 0.8692 = 0.1308.
13Calculate the difference: 0.8692 - 0.1308 = 0.7384.

Answer

The estimated probability is 0.738 (to 3 significant figures).

The continuity correction is vital when approximating a discrete distribution with a continuous one. Always remember to adjust the boundaries by 0.5.

Common mistakes

  • ✗Forgetting to check the conditions (np > 5 and n(1-p) > 5) before using a Normal approximation to the Binomial distribution.
  • ✗Incorrectly applying the continuity correction, especially for inequalities (e.g., using r+0.5 for P(X < r) instead of r-0.5).
  • ✗Confusing standard deviation (σ) with variance (σ²) when using the Normal distribution parameters or standardising.
  • ✗Calculating probabilities for Z < -z using P(Z < z) directly, instead of 1 - P(Z < z) or using calculator functions correctly.
  • ✗Not showing sufficient working, particularly the standardisation step (Z-score calculation) in Normal distribution problems.

Exam tips

  • ★Always state the distribution and its parameters (e.g., X ~ B(10, 0.5) or X ~ N(175, 8²)) at the start of your solution.
  • ★When using Normal approximation, clearly state the mean and variance of the approximating Normal distribution and show the continuity correction applied.
  • ★Use your calculator effectively for Binomial and Normal probabilities, but ensure you write down the formula or the values you are inputting.
  • ★Sketch a Normal curve for probability questions to help visualise the area you need to calculate and to check for sensible answers.

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