Exercises Week 2.docx - Probability Data Tutorial 1 1 Class has 50 students;20 male(M 25 brown-eyed(B For a randomly chosen student what is the range of

# Exercises Week 2.docx - Probability Data Tutorial 1 1 Class...

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Probability & Data Tutorial 1 1. Class has 50 students;20 male (M), 25 brown-eyed (B). For a randomly chosen student what is the range of possible values for P ( M B ) ? 2. Let A, B and C be sets, Use Venn diagrams to show the following: (a) The 2nd distributive law A ∩ ( B C ) = ( A∩ B ) ( A∩ C ) , (b) The 2nd De-Morgan's Law ´ A ∩B = ´ A ´ B. Use the lecture notes and Venn Diagrams to prove the above. 3. Explain from the basis of C r n , why P r n = r !×C r n . 4. A committee consisting of 1 President, 2 Vice-Presidents and 1 Treasurer is selected from a society consisting of 10 members. (a) How many ways can we select the committee? (b) How many ways can we arrange the committee in a row at the front of the stage, if the 2 Vice-President positions are considered equal? (c) How many ways can we arrange the committee around a round table, if the 2 Vice-President positions are considered equal? 5. Use Venn diagrams and the Probability Axioms to show that P ( A B C ) = P ( A ) + P ( B ) + P ( C ) P ( A∩ B ) P ( B∩C ) P ( A∩C ) + P ( A∩ B∩C ) .
6. Show that if A B , then P ( A ) ≤ P ( B ) . 7. What is the probability of getting 3 tails in 5 tosses of a fair coin? Please use a counting method for this question - in a later chapter we will discuss the binomial distribution, where this sort of probability can be calculated by using a formula.

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