Miscellaneous
Last updated at May 9, 2024 by Teachoo
Misc 5 (a) Method 1 Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among the 100 students, what is the probability that you both enter the same sections? n(S) = 100C40 Let A be event that both enter section A n(A) = 98C38 n(S) = 100C60 Let B be event that both enter section B n(B) = 98C58 n(S) = 100C40, n(A) = 98C38 P(A) = (๐(๐ด))/(๐(๐)) = ๐๐๐ช๐๐/๐๐๐๐ช๐๐ = 98C38 รท 100C40 = 98!/38!(98 โ38)! ร 100!/40!(100 โ 40)! = 98!/38!60! รท 100!/40!60! = 98!/38!60! ร 40!60!/100! = (98! ร 40! ร 60!)/(38! ร 60! ร 100!) = (40 ร 39)/(100 ร 99) n(S) = 100C60, n(B) = 98C58 P(B) = (๐(๐ต))/(๐(๐)) = ๐๐๐ช๐๐/๐๐๐๐ช๐๐ = 98C58 รท 100C60 = 98!/58!(98 โ58)! รท 100!/60!(100 โ60)! = 98!/58!40! รท 100!/60!40! = 98!/58!40! ร 60!40!/100! = (98! ร 60! ร 40! )/(58! ร 40! ร100!) = (60 ร 59)/(100 ร 99) Probability that two students enter same section = Probability that both enter Section A + Probability that both enter Section B = P(A) + P(B) = (40 ร 39)/(100 ร 99) + (60 ร 59)/(100 ร 99) = 1560/9900 + 3540/9900 = 156/990 + 354/990 = (156 + 354)/990 = 510/990 = 51/99 = ๐๐/๐๐ Misc 5 (a) - Method 2 Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among the 100 students, what is the probability that you both enter the same sections? Both enter Section A It has 40 students Probability of selecting 2 students out of 40 from a total of 100 students = (40๐ถ_2)/(100๐ถ_2 ) Both enter Section B It has 60 students Probability of selecting 2 students out of 60 from a total of 100 students = (60๐ถ_2)/(100๐ถ_2 ) = ((40!/(2! 38!)))/((100!/(2! 98!)) ) = 40!/(2! 38!) รท 100!/(2! 98!) = 40!/(2! 38!) รท (2! 98!)/100! = (40 ร 39)/(100 ร 99) = ((60!/(2! 58!)))/((100!/(2! 98!)) ) = 60!/(2! 58!) รท 100!/(2! 98!) = 60!/(2! 58!) รท 2!98!/100! = (60 ร 59)/(100 ร 99) Thus, Probability that two students enter same section = Probability that both enter Section A + Probability that both enter Section B = P(A) + P(B) = (40 ร 39)/(100 ร 99) + (60 ร 59)/(100 ร 99) = 1560/9900 + 3540/9900 = 156/990 + 354/990 = (156 + 354)/990 = 510/990 = 51/99 = ๐๐/๐๐ Misc 5 Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among the 100 students, what is the probability that (b) you both enter the different sections? Probability that they enter different section = 1 โ Probability that both enter in same section = 1 โ 17/33 = (33 โ 17)/33 = ๐๐/๐๐