Q27 of 45 Page 1

In a hockey match, both teams A and B scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.

As captain of team A has started to throw the die, he may win in the first throw or third or fifth and so on.


Team B can win in 2nd throw or 4th or 6th and so on


As probability of getting 6 on a die = 1/6 = P(6)


P(6’) represents event of not getting 6 = 5/6


Let P(A) represents the probability that team A wins the game and P(B) represents the probability that team B wins the game.


∴ P(A) = P(6) + P(6’)P(6’)P(6) + P(6’)P(6’)P(6’)P(6’)P(6)+…to ∞


⇒ P(A) =


⇒ P(A) =


Clearly it forms an infinite GP and sum of infinite GP is given by where a is the first term and r is common ratio.


Here a = 1 and r = (5/6)2


∴ P(A) =


Similarly,


P(B) = P(6’)P(6) + P(6’)P(6’)P(6’)P(6) + … to ∞


⇒ P(B) =


⇒ P(B) =


Clearly it forms an infinite GP and sum of infinite GP is given by where a is the first term and r is common ratio.


Here a = 1 and r = (5/6)2


∴ P(B) =


Clearly, P(A) > P(B)


Hence, there is more chance of winning of team A.


∴ Umpire’s decision is unfair. He has favoured team A.


More from this chapter

All 45 →
25

Find the particular solution of the differential equation

(tan–1 y – x) dy = (1 + y2) dx, given that when x = 0, y = 0.

 26

Find the equation of the plane passing through the line of intersection of the planes and whose perpendicular distance from origin is unity.

OR


Find the vector equation of the line passing through the point (1, 2, 3) and parallel to the planes and

28

A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at ₹ 100 and ₹ 120 per unit respectively, how should he use his resources to maximise the total revenue? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

 29

The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.