Mary is a boxer who has never fought professionally before. The chance she wins
ID: 3170779 • Letter: M
Question
Mary is a boxer who has never fought professionally before. The chance she wins her first night is 90%. If she wins then she will fight a more difficult opponent where Mary's chance of winning is 80%. If she loses, she will fight an opponent of the same level so her chance of winning is 90%. This process continues where (i) if she wins she will fight someone at a higher level where her chance of winning reduces by 10% (e.g. if she beats an opponent who she has a chance of beating, her next fight will be against someone who she has a 60% chance of beating); and (ii) if she loses, her next opponent is against someone who as at the same level (e.g. if she loses to someone who she has a 50% chance of beating, she will fight against someone who she has a 50%).
a.) What is the chance Mary has an 80% chance of winning her third fight?
b.) What is the chance Mary wins her third fight?
c.) What is the chance Mary won her first fight, given that she wins her third fight?
d.) Give that the chance of her winning her third fight was 80%, what is the chance she wins her fifth fight?
e.) Let D be the event that Mary won her first fight and E be the eveent hat the chance Mary won her third fight was 80%? Are D and E independent? If you changed E to the event that the chance of Mary won third fight was 90% would D and E be independent?
Explanation / Answer
a) Chance Mary has an 80% chance of winning her tihrd fight.
This can only occur if she has one exactly 1 of her first 2 fights because the net reduction in chance of winning is only 10% from 90% to 80% after 2 fights.
So 2 possibilities are WL and LW
P( WL + LW) = 0.9*(1-0.8) + ( 1-0.9)*(0.9) = 0.9*0.2 + 0.1*0.9 = 0.18+ 0.09 = 0.27
b) Chance Mary wins her third fight, This could be done in many ways: WWW, WLW, LWW, LLW
P(WWW+ WLW+ LWW+ LLW) = 0.9*0.8*0.7 + 0.9*0.2*0.8+ 0.1*0.9*0.8+ 0.1*0.1*0.9
= 0.504+ 0.144 + 0.072+ 0.009 = 0.729
c) Chance that Mary wins first fight, given that she wins her third fight
P ( WWW+WLW) / P(WWW+ WLW+ LWW+ LLW) = (0.504 + 0.144) / 0.729 = 0.889
d) Chance of winning 3rd fight =80%
possible courses till 5th fight: WW, WL, LW, LL
Therefore chance of winning in 5th fight:
0.56* 60% + (0.24+ 0.16) * 70% + 0.04* 80%
= 33.6% + 28% + 3.2% = 64.8%
Therefore 64.8% is the required answer
Course of 3rd and 4th fight Probability of that occuring Chance of winning 5th fight WW 0.8*0.7 = 0.56 60% WL 0.8*0.3 = 0.24 70% LW 0.2*0.8= 0.16 70% LL 0.2*0.2 = 0.04 80%