Problem 7. [10 points] Bubba and Stu are shooting at a road sign. They take shots in this order Bubba, Stu, Stu, Bubba, Bubba, Stu, Stu, Bubba, Bubba, Stu, Stu, etc With each shot: Bubba hits the sign with probability 2/5 Stu hits the sign with probability 1/4 What is the probability that Bubba hits the sign before Stu? Assume that hits occur mu tually independently. You must give a closed-form answer to receive full credit Solution Pr(bubba hits first)==+ 3/3)223/3232 43-4 55(4)55 (3)()3+:()() 2 2-2 2-52-52-5 () 6 3 252(4(5 68 255 16 26825 52= 592(1625 515 400 216 5+15(1-81/400 21681 5+15319 214 319� � � � Final Exam 11 Problem 7. [10 points] Bubba and Stu are shooting at a road sign. They take shots in this order: Bubba, Stu, Stu, Bubba, Bubba, Stu, Stu, Bubba, Bubba, Stu, Stu, etc. With each shot: • Bubba hits the sign with probability 2/5. • Stu hits the sign with probability 1/4. What is the probability that Bubba hits the sign before Stu? Assume that hits occur mu￾tually independently. You must give a closed­form answer to receive full credit. Solution. 2 Pr(Bubba hits first) = +5 � �2 � �2 3 3 2 3 3 3 2 + + 5 4 5 5 4 5 5 � �2 � �2 � �2 � �2 � �2 � �2 3 3 3 3 2 3 3 3 3 3 2 + + 5 4 5 4 5 5 4 5 4 5 5 . . . 2 3 2 �∞ �� �2i � �2i−2 � �2i � �2i−1 � 3 3 3 3 = + + 5 5 5 4 5 4 5 i=1 ∞ � �2i−2 � � 2 6 � 3 �2i �3 3 = + 1 + 5 25 4 5 5 i=1 ∞ � �i−1 2 6 8 � 9 �i � 9 = +5 25 5 16 25 i=1 ∞ 2 6 8 25 � 9 9 �i = +5 25 5 9 16 25 i=1 ∞ 2 16 � 81 �i = +5 15 400 i=1 2 16 1 = 5 + 15 1 − 81/400 − 1 2 16 81 = +5 15 319 214 = 319