Ex 14.1, 6 - Chapter 16 Class 11 Probability (Important Question)

Transcript

Ex 14.1, 6 Two dice are thrown. The events A, B and C are as follows: A: getting an even number on the first die. B: getting an odd number on the first die. C: getting the sum of the numbers on the dice ≤ 5 Describe the events If 2 dies are thrown then possible outcomes are 1, 2, 3, 4, 5, 6 on both dies S = {█("(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)," @"(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)," @"(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)," @"(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)," @"(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)," @"(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)" )} A: getting an even number on the first die A = {█("(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)," @"(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), " @"(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)" )} B: getting an odd number on the first die B = {█("(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)," @" (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)," @" (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) " )} C: getting the sum of the numbers on the dice ≤ 5 "(1, 1), (1, 2), (1, 3), (1, 4), " "(2, 1), (2, 2), (2, 3)," Ex 14.1, 6 A’ S = {█("(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)," @"(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)," @"(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)," @"(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)," @"(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)," @"(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)" )} A = {█("(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)," @"(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), " @"(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)" )} A’ = S – A A’ = {█("(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)," @" (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)," @"(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), " )} = Getting odd number on the first die = B Ex 14.1, 6 (ii) not B S = {█("(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)," @" (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), " @"(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)," @"(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)," @" (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)," @"(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}" )} B = {█("(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)," @" (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)," @" (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) " )} not B = S – B not B = {█("(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)," @"(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), " @"(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)" )} = Getting even number of the first die = A Ex 14.1, 6 (iii) A or B A = {█("(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)," @"(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), " @"(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)" )} B = {█(█((1, 1),(1, 2),(1, 3),(1, 4),(1, 5),(1, 6),@(3, 1),(3, 2),(3, 3)," " (3, 4),(3, 5),(3, 6) )@(5, 1),(5, 2),(5, 3),(5, 4),(5, 5),(5, 6) )} A or B = A ∪ B = {█("(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)," @" (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), " @"(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)," @"(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)," @" (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)," @"(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)" )} = S Ex 14.1, 6 (iv) A and B A ={█("(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)," @"(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), " @"(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)" )} B = {█(█((1, 1),(1, 2),(1, 3),(1, 4),(1, 5),(1, 6),@(3, 1),(3, 2),(3, 3)," " (3, 4),(3, 5),(3, 6) )@(5, 1),(5, 2),(5, 3),(5, 4),(5, 5),(5, 6) )} A and B = A ∩ B = 𝛟 Ex 14.1, 6 (v) A but not C A = {█("(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)," @"(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), " @"(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)" )} C = {█((1, 1), (1, 2), (1, 3), (1, 4),@(2, 1), (2, 2), (2, 3),@(3, 1), (3, 2), (4, 1))} A but not C = A – C = {█(█((2, 4),(2, 5),(2, 6),@(4, 2),(4, 3),(4, 4),(4, 5),(4, 6)"," )@█((6, 1),(6, 2),(6, 3),(6, 4),(6, 5),(6 6) ))} Ex 14.1, 6 (vi) B or C B = {█(█((1, 1),(1, 2),(1, 3),(1, 4),(1, 5),(1, 6),@(3, 1),(3, 2),(3, 3)," " (3, 4),(3, 5),(3, 6) )@(5, 1),(5, 2),(5, 3),(5, 4),(5, 5),(5, 6) ),} C = {█((1, 1), (1, 2), (1, 3), (1, 4),@(2, 1), (2, 2), (2, 3),@(3, 1), (3, 2), (4, 1))} B or C = B ∪ C = {█(█((1, 1),(1, 2),(1, 3),(1, 4),(1, 5),(1, 6),@(2, 1), (2, 2), (2, 3),@(3, 1),(3, 2),(3, 3)," " (3, 4),(3, 5),(3, 6),@(4, 1),)@(5, 1),(5, 2),(5, 3),(5, 4),(5, 5),(5, 6),)} Ex 14.1, 6 (vii) B and C B = {█((1, 1),(1, 2),(1, 3),(1, 4),(1, 5),(1, 6)", " @█((3, 1),(3, 2),(3, 3), (3, 4),(3, 5),(3, 6)@(5, 1),(5, 2),(5, 3),(5, 4),(5, 5),(5, 6) ))} C = {█((1, 1), (1, 2), (1, 3), (1, 4),@(2, 1), (2, 2), (2, 3),@(3, 1), (3, 2), (4, 1))} B and C = B ∩ C = {(1, 1),(1, 2),(1, 3),(1, 4), (3, 1),(3, 2),} Ex 14.1, 6 (viii) A ∩ B’ ∩ C’ We know A & B’ (calculated in part(ii)) Finding C’ S = {█("(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)," @"(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)," @"(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)," @"(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)," @"(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)," @"(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)" )} C = {█((1, 1), (1, 2), (1, 3), (1, 4),@(2, 1), (2, 2), (2, 3),@(3, 1), (3, 2), (4, 1))} C’ = S – C = {█("(1, 5), (1, 6)," @"(2, 4), (2, 5), (2, 6)," @ "(3, 3), (3, 4), (3, 5), (3, 6)," @"(4, 2), (4, 3), (4, 4), (4, 5), (4, 6)," @"(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)," @"(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)" )} Also, A = {█((2, 1),(2, 2),(2, 3),(2, 4),(2, 5),(2, 6),@█((4, 1),(4, 2),(4, 3),(4, 4),(4, 5),(4, 6),@(6, 1),(6, 2),(6, 3),(6, 4),(6, 5),(6, 6) ))} B’ = {█((2, 1),(2, 2),(2, 3),(2, 4),(2, 5),(2, 6),@█((4, 1),(4, 2),(4, 3),(4, 4),(4, 5),(4, 6),@(6, 1),(6, 2),(6, 3),(6, 4),(6, 5),(6, 6) ))} Thus, A ∩ B’ ∩ C’ = {█((2, 4),(2, 5),(2, 6)"," @" " (4, 2),(4, 3),(4, 4),(4, 5),(4, 6),@(6, 1),(6, 2),(6, 3),(6, 4),(6, 5),(6 6) )}