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    IBMediumProbability10–12

    Three Languages in One Venn Diagram

    Três Línguas num Diagrama de Venn

    Of 120 students, 60 study Spanish, 50 study French and 40 study German. Also 25 study Spanish and French, 18 study Spanish and German, 15 study French and German, and 8 study all three. Find how many study none of the three languages, and the probability that a student chosen at random studies exactly one language.

    For the union, the three-set rule adds the single totals, removes the pairs and puts the triple back. For "exactly one", do not work from the circle totals: fill the diagram from the centre outwards, starting with the students in all three.

    Solution

    Step 1 of 16

    1. 1.Union by inclusion–exclusion: