A and B take part in a duel. A can strike with an accuracy of 0.6. B c

Question

A and B take part in a duel. A can strike with an accuracy of 0.6. B can strike with an accuracy of 0.8. A has the first shot, post which they strike alternately. What is the probability that A wins the duel?

A. 11/17
B. 15/23
C. 2/3
D. 7/10
E. 7/9

Are You Up For the Challenge: 700 Level Questions

A. 11/17
B. 15/23
C. 2/3
D. 7/10
E. 7/9

Are You Up For the Challenge: 700 Level Questions

nick1816 · Accepted Answer

Probability that A wins the duel on his first turn  = 0.6

Probability that A wins the duel on his second turn  = 0.4 * 0.2 *0.6  (A and B miss on their first turn)

Probability that A wins the duel on his third turn  = 0.4 * 0.2 *0.4 * 0.2*0.6= (0.4 * 0.2)^2 * 0.6  (A and B miss on their first and second turn)

and so on

Probability that A wins the duel on his nth turn =  (0.4 * 0.2)^{n-1} * 0.6

the probability that A wins the duel  = 0.6 + (0.4 * 0.2) * 0.6+ (0.4 * 0.2)^2 * 0.6+....... upto infinite turns

Sum of infinite GP  = \frac{a}{1-r}  {a=first term; r=common ratio}

the probability that A wins the duel  = \frac{0.6}{  }= \frac{60}{92} = \frac{15}{23}

GMATinsight · Answer

Probability of A striking = 0.6
i.e. Probability of A failing to strike = 1-0.6 = 0.4

Probability of B striking = 0.8
i.e. Probability of B failing to strike = 1-0.8 = 0.2

Case 1: A wins in first shot

Probability = 0.6

Case 2: A wins in second shot i.,e. sequence of strike is  A Fail, B fails, A strikes

Probability  = 0.4*0.2*0.6

Case 3: A wins in third shot i.,e. sequence of strike is  A Fail, B fails, A Fail, B fails, A strikes

Probability  = 0.4*0.2*0.4*0.2*0.6

and so on

Final Probability = Sum of all cases  = 0.6*  = 0.6*

CONCEPT : Sum of terms of an Infinite GP with common ratio r such that  -1 < r < 1 = \frac{a}{(1-r)}

Final Probability  = 0.6*  = 0.6*(\frac{1}{(1-0.08) }= 0.6*(\frac{1}{0.92}) = 60/92 = 15/23

GMATWhizTeam · Answer

Solution 
 • Let us assume that a denotes probability of A to strike the target and  a ̅   denotes the probability that A doesn’t strike the target.
 o  a = 0.6 
o  a ̅ = (1-0.6) = 0.4  
• Similarly, let us assume that b denotes probability of B to strike the target and  b ̅   denotes the probability that B doesn’t strike the target.
 o  b = 0.8 
o  b ̅ = (1-0.8) = 0.2  
• Now, possible ways in which A can win = A strike the target on his first shot OR A strike the target on his second shot OR A strike the target on his third shot OR .....and so on.
• Since, A and B are striking alternately.
 o So, the probability of A to win(Let’s say P)  = a + a ̅*b ̅ *a + a ̅*b ̅ *a ̅*b ̅ *a +a ̅*b ̅ *a ̅*b ̅ *a ̅*b ̅ *a + …….  And so on
o  P = 0.6 + 0.4*0.2*0.6 + 0.4*0.2*0.4*0.2*0.6 + 0.4*0.2*0.4*0.2*0.4*0.2*0.6 + …………  and so on.
o  P = 0.6    
• We can observe that the terms in the bracket   forms a G.P. with first term as 1 and common ratio as 0.08.
• Therefore,  P = 0.6* \frac{1}{1- 0.08} = \frac{0.6}{0.92} = \frac{60}{92} = \frac{15}{23}  
Thus, the correct answer is  Option B.

sanjitscorps18 · Answer

Hi  Bunuel ,

When we say A wins the duel, I think we have to give equal chances to A and B and not set this up disproportionately. 
If we are not supposed to give B his second turn, then why even give him the opportunity in the first turn.

IMO the sequences should be (bold parts should be needed)
Game finishes in 1 turn -> A passes *  B fails 
Game finishes in 2 turns ->   * A passes *  B fails 
Game finishes in 2 turns ->   *   * A passes *  B fails  and so on

Hence, we have
A's Win requirement = A passes * B fails =   = 3/25
All possible reasons for A and B to remain equal at the end of the round =   =   = 2/25 + 12/25 = 14/25

Final probability = 3/25 + 14/25 * 3/25 + (14/25)^2 * 3/25 + (14/25)^3 * 3/25 + ......
=> 3/25  
=> 3/25 * 1/ 
=> 3/25 * 25/11
IMO Probability should be 3/11

I'll await your thoughts.

Bunuel · Answer

You’re mixing up alternating turns with simultaneous shots. A and B don’t shoot in the same round. A fires first; if he misses, then B fires. The duel ends as soon as one hits (meaning the hit kills the other).

sanjitscorps18 · Answer

I think I took duel in some other sense (like a game) rather than some life and death moment. Thanks.

A and B take part in a duel. A can strike with an accuracy of 0.6. B c

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A and B take part in a duel. A can strike with an accuracy of 0.6. B c

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