Question
Tom and Jerry throw alternatively a pair of dice. Tom
wins if he throws 8 before Jerry throws 7 and Jerry wins if he throws 7 before Tom throws 8. Find their respective chance of winning, if Tom begins.Solution
8 can be thrown with a pair of dice in the following ways (2,6), (6,2), (4,4), (5,3), (3,5) So, Probability of throwing ‘8’ = 5/36 And Probability of not throwing ‘8’ = 31/36 And 7 can be thrown with a pair of dice in the following ways (1,6), (6,1), (2,5), (5,2), (3,4), (4,3) So, the probability of throwing a ‘7’ = 6/36= 1/6 And Probability of not throwing ‘7’ = 5/6 Let E1 be the event of the throwing a ‘8’ in a single throw of a pair of dice and E2 be the event of throwing a ‘7’ in a single throw of a pair of Dice Then P(E1) = 5/36 , P(E2) = 1/6 And P(¯E1)= 31/36, P(¯E2) = 5/6 Tom wins if he throws ‘8’ in first or third or fifth............ throws. Probability of Tom throwing a ‘8’ in First throw = P(E1) = 5/36 And Probability of Tom throwing a ‘8’ in third throw = P(¯E1 ∩ ¯E2 ∩ E1 ) = P(¯E1) × P(¯E2) × P(E1) = 31/36 × 5/6 ×5/36 Similarly, Probability of Tom throwing a ‘8’ in fifth throw = P(¯E1) × P(¯E2) × P(¯E1) × P(¯E2) × P(E1) = (31/36)^2×(5/6)² ×5/36 Hence, Probability of winning of Tom = P[E1∪(¯E1 ∩ ¯E2 ∩ E1)∪ (¯E1∩ ¯E2 ∩ ¯E1∩¯E2∩ E1)∪……. ] = P[E1+(¯E1 ∩ ¯E2 ∩ E1)+ (¯E1∩ ¯E2 ∩ ¯E1∩¯E2∩ E1)+⋯…. ] Sum of n terms of Geometric Progression , if r < 1 = a/(1-r) ∴ Probability of winning of Tom = (5/36)/(1-(31/36 × 5/6)) = 30/61 Thus, Probability of winning of Jerry = 1 - 30/61 = 31/61
World Hearing Day is observed every year on which date?
Which of the following firm has recently launches 360-degree panaromic street view 'Mappls RealView'?
The Invest Rajasthan Summit,2022 which was supposed to be held on 24th and 25th January, 2022 in Jaipur was postponed due to emerging cases of Covid 19,...
What was the primary reason cited by Russia for formally withdrawing from the 1990 Treaty on Conventional Armed Forces in Europe (CFE)?
Who partnered with the State Bank of India (SBI) for the 'Grow with the Trees' drive in Srinagar?
'Shell India' has appointed whom as its new chairperson?
Which Indian film was selected for the Cannes Film Festival in the ‘La Cinef’ category in 2024?
The first common forum of dispute resolution between the Centre and states,the Goods and Services Tax Appellate Tribunal (GSTAT) will be having how many...
The union cabinet has approved and finalized the development of the new Dholera greenfield airport in which of the following states?
Atal Vayo Abhyuday Yojana (AVYAY) is aimed at supporting which segment of the population?