A, B and C Throw a Pair of Dice in that Order Alternatively Till One of Them Gets a Total of 9 and Wins the Game. Find Their Respective Probabilities of Winning, If a Starts First. - Mathematics

Advertisements
Advertisements
Sum

A, B and C throw a pair of dice in that order alternatively till one of them gets a total of 9 and wins the game. Find their respective probabilities of winning, if A starts first.

Advertisements

Solution

Let E: getting a total of 9 

∴ `"E" = {(3,6),(4,5),(5,4),(6,3)}`

So, clearly 4 1 probability of winning = P(E) = `(4)/(36) = (1)/(9), "P"(bar"E") = (8)/(9)`


As A starts the game so, he may win in 1st, 4th, 7th, ... trials.

`"P"("A wins") = "P"(bar"E") + "P"(bar"E")"P"(bar"E")"P"(bar"E")"P"(bar"E")+ "P"(bar"E")"P"(bar"E")"P"(bar"E")"P"(bar"E")"P"(bar"E")"P"(bar"E")"P"(bar"E") + ...`


⇒ `"P"("A wins") = (1)/(9) + (8/9)^3 xx (1)/(9) (8/9)^6 xx (1)/(9)+...`


∴  `"P"("A wins") = ((1)/(9))/((1- 512)/(729))`


= `(81)/(217)`.


Now B may win in 2nd, 5th, 8th,...trials.


∴  `"P"("A wins") = "P"(bar"E")"P"(bar"E") + "P"(bar"E")"P"(bar"E")"P"(bar"E")"P"(bar"E")"P"(bar"E")+ "P"(bar"E")"P"(bar"E")"P"(bar"E")"P"(bar"E")"P"(bar"E")"P"(bar"E")"P"(bar"E")"P"(bar"E")+...`


⇒ `"P"("A wins") = (8)/(9) xx (1)/(9) + (8/9)^4 xx (1)/(9) + (8/9)^7 xx (1)/(9) +...`

 

∴  `"P"("A wins") = ((8)/(81))/(1 - (821)/(729))`


= `(72)/(217)`


And, finally 

`"P"("C wins") = 1-["P"("A wins")+"P"("B wins")]`


= `1 - [(81)/(217 + (72)/(217)]]`


= `(64)/(217)`.

Concept: Probability Examples and Solutions
  Is there an error in this question or solution?
2015-2016 (March) All India Set 1 E

RELATED QUESTIONS

A and B throw a pair of dice alternately, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first


In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at random from this set and tossed five times. If all the five times, the result was heads, find the probability that the selected coin had heads on both the sides.


A coin is tossed three times, if head occurs on first two tosses, find the probability of getting head on third toss.


A die is thrown three times, find the probability that 4 appears on the third toss if it is given that 6 and 5 appear respectively on first two tosses.


If P (A) = 0.4, P (B) = 0.3 and P (B/A) = 0.5, find P (A ∩ B) and P (A/B).

 

If A and B are two events such that P (A) = \[\frac{1}{3},\] P (B) = \[\frac{1}{5}\] and P (A ∪ B) = \[\frac{11}{30}\] , find P (A/B) and P (B/A).

 
 
 

A couple has two children. Find the probability that both the children are (i) males, if it is known that at least one of the children is male. (ii) females, if it is known that the elder child is a female.


If A and B are events such that P (A) = 0.6, P (B) = 0.3 and P (A ∩ B) = 0.2, find P (A/B) and P (B/A).


A dice is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?


A pair of dice is thrown. Find the probability of getting 7 as the sum if it is known that the second die always exhibits a prime number.


A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die.


A die is thrown twice and the sum of the numbers appearing is observed to be 8. What is the conditional probability that the number 5 has appeared at least once?


The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3, and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.


Given two independent events A and B such that P (A) = 0.3 and P (B) = `0.6. Find P (A ∩ overlineB ) `.


Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find P (A ∪ B).


If A and B are two independent events such that P (A ∪ B) = 0.60 and P (A) = 0.2, find P(B).


A die is thrown thrice. Find the probability of getting an odd number at least once.

 

An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting 2 red balls.  


An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting one red and one blue ball.


Arun and Tarun appeared for an interview for two vacancies. The probability of Arun's selection is 1/4 and that to Tarun's rejection is 2/3. Find the probability that at least one of them will be selected.


Three persons ABC throw a die in succession till one gets a 'six' and wins the game. Find their respective probabilities of winning.


An urn contains 7 red and 4 blue balls. Two balls are drawn at random with replacement. Find the probability of getting
(i) 2 red balls
(ii) 2 blue balls
(iii) One red and one blue ball.


A card is drawn from a well-shuffled deck of 52 cards. The outcome is noted, the card is replaced and the deck reshuffled. Another card is then drawn from the deck.
(i) What is the probability that both the cards are of the same suit?
(ii) What is the probability that the first card is an ace and the second card is a red queen?


One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is white.


An urn contains 10 white and 3 black balls. Another urn contains 3 white and 5 black balls. Two are drawn from first urn and put into the second urn and then a ball is drawn from the latter. Find the probability that its is a white ball.


Write the probability that a number selected at random from the set of first 100 natural numbers is a cube.

 

If ABC are mutually exclusive and exhaustive events associated to a random experiment, then write the value of P (A) + P (B) + P (C).


A and B draw two cards each, one after another, from a pack of well-shuffled pack of 52 cards. The probability that all the four cards drawn are of the same suit is


The probability that a leap year will have 53 Fridays or 53 Saturdays is


Five persons entered the lift cabin on the ground floor of an 8 floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first, then the probability of all 5 persons leaving at different floors is


A bag contains 5 brown and 4 white socks. A man pulls out two socks. The probability that these are of the same colour is


Choose the correct alternative in the following question:
If A and B are two events associated to a random experiment such that \[P\left( A \cap B \right) = \frac{7}{10} \text{ and } P\left( B \right) = \frac{17}{20}\] , then P(A|B) = 


Mark the correct alternative in the following question:

\[ \text{ If }  P\left( B \right) = \frac{3}{5}, P\left( A|B \right) = \frac{1}{2} \text{ and }  P\left( \overline{A \cup B }\right) = \frac{4}{5}, \text{ then }  P\left( \overline{ A } \cup B \right) + P\left( A \cup B \right) = \]


Mark the correct alternative in the following question:

\[\text{ If A and B are two events such that} P\left( A \right) \neq 0 \text{ and }  P\left( B \right) \neq 1,\text{ then } P\left( \overline{ A }|\overline{ B }\right) = \]


Mark the correct alternative in the following question:

\[\text{ If the events A and B are independent, then }  P\left( A \cap B \right) \text{ is equal to } \]


Mark the correct alternative in the following question: A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement, then the probability that exactly two of the three balls were red, the first ball being red, is


Mark the correct alternative in the following question: 

\[\text{ If A and B are such that } P\left( A \cup B \right) = \frac{5}{9} \text{ and } P\left( \overline{A} \cup \overline{B} \right) = \frac{2}{3}, \text{ then } P\left( A \right) + P\left( B \right) = \]


Share
Notifications



      Forgot password?
Use app×