Theory question 106
SetsIn a class of 40 students, 18 passed Mathematics, 19 passed Accounts, 16 passed Economics, 5 Mathematics and Accounts only, 6 Mathematics only, 9 Accounts only, 2 Accounts and Economics only. If each student offered at least one of the subjects, (a) how many students failed in all the subjects? (b) find the percentage number who failed in at least one of Economics and Mathematics; (c) calculate the probability that a student at random failed in Accounts.
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Let x passed all three. Accounts: 9 + 5 + 2 + x = 19, so x = 3.
Mathematics and Economics only = 18 − 6 − 5 − 3 = 4. Economics only = 16 − 2 − 3 − 4 = 7.
(a) Passed at least one = 6 + 9 + 7 + 5 + 2 + 4 + 3 = 36, so 40 − 36 = 4 failed all.
(b) Passed both Economics and Mathematics = 4 + 3 = 7, so 40 − 7 = 33 failed at least one of them: 33/40 × 100% = 82.5%.
(If read as failing both: 40 − n(M ∪ E) = 40 − 27 = 13, i.e. 32.5%.)
(c) Failed Accounts = 40 − 19 = 21; probability = 21/40.
Fill in the Venn diagram from the 'only' numbers first, then use the totals.