Fraunhofer Institute for Industrial Mathematics ITWMPodcast
»Streuspanne-Lexicon« – B for Binomial Distribution
Today's »Streuspanne-Lexicon« entry is about what you need the binomial distribution for and what it actually is. As usual: explained briefly and clearly – in under five minutes.
Podcast »Streuspanne – Statistics and Their Curiosities«
»One figure in every seventh egg!« – In the example of the encyclopaedia article, the »Streuspanne«-team is interested in how often a Happy Hippo figure can be found in a surprise egg if someone were to buy an egg every day for a month. Here, the individual probability of success »p« would be exactly one seventh, the number of repetitions »N« would be 30 days and »k« is the random result, namely the number of successes. The binomial distribution determines how often you can expect which value of k.
In addition to surprise eggs, it can also be used to calculate other probabilities (e.g. for a coin or dice roll) and make predictions about complex processes such as the spread of diseases.
The formula for the binomial distribution is (N over k) times p^k times (1-p)^(N-k).
At first glance this looks complicated, but with a little reading help the formula loses its terror.
p^k is easy to understand: If I want to have k successes in N repetitions, then the success must occur k times, which happens with probability p. In other words, simply the probability p multiplied by itself k times.
Similarly, it is easy to understand why (1-p)^(N-k) is in the formula. Because with k successes, I then have to reckon with exactly N-k failures, and each individual failure has a probability of 1-p.
The last component (N over k) is somewhat more abstract. This is the so-called binomial coefficient. What this means mathematically would be too much for a dictionary entry, but it simply counts the number of ways in which k successes can be accommodated in N attempts. Heavy words, but with a simple meaning.
But What Does This Binomial Distribution Look Like?
As N increases, the binomial distribution becomes closer and closer to a normal distribution. The number of hits is distributed symmetrically to the left and right of the expected value, i.e. exactly in the middle at p=1/2. The number of hits is highest exactly at this expected value and decreases in both directions.
In a nutshell: The binomial distribution counts the number of successes in a fixed number of repetitions. You can find out what this means for the surprise eggs and the figures in the »Streuspanne-Lexicon«.
Have you discovered a strange statistic in the media and want us to talk about it in the podcast? Or have you noticed a mathematical number game or thought experiment? Then get in touch with us at presse(at)itwm.fraunhofer.de. Suggestions for further lexicon entries are also welcome.
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