☰ Contents · Mathematics

Complementary events, addition and multiplication

Lessons 34–35 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 11, Parts 1 and 2, 1st edition. ZAMIN NASHR, Tashkent, 2018
34

Complementary events, operations on events and Euler–Venn diagrams

Textbook, Part 2: pp. 78–82
GoalLearn to describe operations on events (sum, product, difference) and complementary events with Euler–Venn diagrams and to use P(Ā) = 1 − P(A).
New words
complementary event · qarama-qarshi hodisamutually exclusive · birgalikda bo‘lmaganEuler–Venn diagram · Eyler–Venn diagrammasielementary event · elementar hodisa
Explanation

If we draw the set U of elementary events as a region, each event is a part of it (an Euler–Venn diagram). The product AB = A ∩ B means both occur together; the sum A + B = A ∪ B means at least one occurs; the difference A − B means A occurs but B does not. The event that A does not occur, Ā, is the complementary event: A + Ā = U and AĀ = V. If AB = V the events A and B are mutually exclusive (cannot occur at once). For probability: P(A) ≥ 0, P(U) = 1, and for mutually exclusive events P(A + B) = P(A) + P(B). From this P(Ā) = 1 − P(A) follows; to find the probability of “at least one”, it is convenient to compute “none” and subtract from 1. In general P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Worked examples
A die is tossed: A = {even} = {2, 4, 6}, B = {greater than 3} = {4, 5, 6}. AB = {4, 6}, A + B = {2, 4, 5, 6}, A − B = {2}, Ā = {1, 3, 5}, P(Ā) = 1 − 1/2 = 1/2. At least one head in 4 coin tosses: 1 − P(none) = 1 − (1/2)⁴ = 15/16.
Of 30 students, 18 do football (F), 12 do chess (C), and 5 do both. In the diagram: football only 13, chess only 7, both 5, neither 30 − 25 = 5. P(F ∪ C) = (18 + 12 − 5)/30 = 25/30 = 5/6; P(neither) = 1 − 5/6 = 1/6.
Class activity

“Circles”: draw two overlapping circles and ask classmates who comes to school by bike (B) and who on foot (W) (some do both). Write the counts in the diagram and find P(B ∪ W), P(B only) and P(neither).

Practice
1
If P(A) = 0.35, find P(Ā).
2
A die is tossed, A = {1, 2, 3}, B = {3, 4}. Find P(A ∪ B).
3
A coin is tossed 4 times. What is the probability of at least one tail?
4
Why are mutually exclusive events not always complementary?