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The birthday experiment is to draw a random sample of size n with replacement from a population of size N. In the applet, the population consists of N balls numbered from 1 to N. We are interested in the event that there is at least one duplication in the sample; the indicator variable of this event is denoted I. On each update, the sample of balls is shown in the first graph; a ball that duplicates a previously chosen ball is shown in red, and otherwise is shown in green. The value of I is recorded on each update in the first table. The discrete probability density function of I is shown in blue in the second graph and is recorded in the second table. On each update, the empirical probability density function of I is shown in red in the second graph and is recorded in the second table. The parameters N and n can be varied with scroll bars.

Software/Applets used on this page



the ratio of the mass of an object to its volume


any set of possible outcomes of a statistical experiment


A rule that connects one value in one set with one and only one value in another set.


A diagram showing a relationship between two variables.
The diagram shows a vertical y axis and a horizontal x axis.


A collection, group or set of individuals or objects.


A collection of sampling units drawn from a sampling frame.


A value which is unknown and free to vary, often denoted by x or y.

Full Glossary List

This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AQA A-Level (UK - Pre-2017)S1ProbabilityExperimental-
AQA AS Maths 2017StatisticsProbabilityExperimental Probabilities-
AQA AS/A2 Maths 2017StatisticsProbabilityExperimental Probabilities-
CBSE XII (India)ProbabilityProbabilityRepeated independent (Bernoulli) trials and binomial distribution-
CBSE XI (India)Statistics and ProbabilityProbabilityRandom experiments, outcomes, sample spaces (set representation)-
CCEA A-Level (NI)S1ProbabilityExperimental-
CIE A-Level (UK)S1ProbabilityExperimental-
Edexcel A-Level (UK - Pre-2017)S1ProbabilityExperimental-
Edexcel AS Maths 2017StatisticsProbabilityExperimental Probabilities-
Edexcel AS/A2 Maths 2017StatisticsProbabilityExperimental Probabilities-
I.B. Higher Level5ProbabilityExperimental-
I.B. Standard Level5ProbabilityExperimental-
Methods (UK)M15ProbabilityExperimental-
I.B. (MSSL)3ProbabilityExperimental-
OCR A-Level (UK - Pre-2017)S1ProbabilityExperimental-
OCR AS Maths 2017StatisticsProbabilityExperimental Probabilities-
OCR MEI AS Maths 2017StatisticsProbabilityExperimental Probabilities-
OCR-MEI A-Level (UK - Pre-2017)S1ProbabilityExperimental-
Pre-Calculus (US)E4ProbabilityExperimental-
Pre-U A-Level (UK)ProbProbabilityExperimental-
Scottish (Highers + Advanced)HSProbabilityExperimental-
Scottish HighersSProbabilityExperimental-
Universal (all site questions)PProbabilityExperimental-
WJEC A-Level (Wales)S1ProbabilityExperimental-