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Consider the following problem: In a bullet firing competition $A$ can hit $4$ out of $5$ bullet in target, similar chances of $B$ is $3$ out of $4$ and for $C$ is $2$ out of $3$.They fire simultaneously and exactly two out of them makes correct hit.What is the probability that $C$ has missed the target?
Let $X$: $A$ hits the target, $Y$: $B$ hits the target,$Z$: $C$ hits the target.
Solution $1$: The required probability is ${P(XYZ^c ) over P(XYZ^c )+P(YZX^c )+P(ZXY^c ) }$=${{4 over 5}{3 over 4}{1 over 3} over {4 over 5}{3 over 4}{1 over 3}+{3 over 4}{2 over 3}{1 over 5}+{2 over 3}{4 over 5}{1 over 4}}$=${6 over 13}$.



Solution $2$: The required probability is $P(XYZ^c )={1 over 5}$.



So, my question is which of the solution is correct, and why the other one is incorrect, that is what is the question that the other solution makes that answer$?$






probability-theory







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  • The probability that only C misses given that two out of three hit, is the same at the probability that only C misses.
    – Math-fun
    Jan 4 at 12:41
















0














Consider the following problem: In a bullet firing competition $A$ can hit $4$ out of $5$ bullet in target, similar chances of $B$ is $3$ out of $4$ and for $C$ is $2$ out of $3$.They fire simultaneously and exactly two out of them makes correct hit.What is the probability that $C$ has missed the target?
Let $X$: $A$ hits the target, $Y$: $B$ hits the target,$Z$: $C$ hits the target.
Solution $1$: The required probability is ${P(XYZ^c ) over P(XYZ^c )+P(YZX^c )+P(ZXY^c ) }$=${{4 over 5}{3 over 4}{1 over 3} over {4 over 5}{3 over 4}{1 over 3}+{3 over 4}{2 over 3}{1 over 5}+{2 over 3}{4 over 5}{1 over 4}}$=${6 over 13}$.



Solution $2$: The required probability is $P(XYZ^c )={1 over 5}$.



So, my question is which of the solution is correct, and why the other one is incorrect, that is what is the question that the other solution makes that answer$?$






probability-theory







share|cite|improve this question






















  • The probability that only C misses given that two out of three hit, is the same at the probability that only C misses.
    – Math-fun
    Jan 4 at 12:41














0












0








0


1





Consider the following problem: In a bullet firing competition $A$ can hit $4$ out of $5$ bullet in target, similar chances of $B$ is $3$ out of $4$ and for $C$ is $2$ out of $3$.They fire simultaneously and exactly two out of them makes correct hit.What is the probability that $C$ has missed the target?
Let $X$: $A$ hits the target, $Y$: $B$ hits the target,$Z$: $C$ hits the target.
Solution $1$: The required probability is ${P(XYZ^c ) over P(XYZ^c )+P(YZX^c )+P(ZXY^c ) }$=${{4 over 5}{3 over 4}{1 over 3} over {4 over 5}{3 over 4}{1 over 3}+{3 over 4}{2 over 3}{1 over 5}+{2 over 3}{4 over 5}{1 over 4}}$=${6 over 13}$.



Solution $2$: The required probability is $P(XYZ^c )={1 over 5}$.



So, my question is which of the solution is correct, and why the other one is incorrect, that is what is the question that the other solution makes that answer$?$






probability-theory







share|cite|improve this question













Consider the following problem: In a bullet firing competition $A$ can hit $4$ out of $5$ bullet in target, similar chances of $B$ is $3$ out of $4$ and for $C$ is $2$ out of $3$.They fire simultaneously and exactly two out of them makes correct hit.What is the probability that $C$ has missed the target?
Let $X$: $A$ hits the target, $Y$: $B$ hits the target,$Z$: $C$ hits the target.
Solution $1$: The required probability is ${P(XYZ^c ) over P(XYZ^c )+P(YZX^c )+P(ZXY^c ) }$=${{4 over 5}{3 over 4}{1 over 3} over {4 over 5}{3 over 4}{1 over 3}+{3 over 4}{2 over 3}{1 over 5}+{2 over 3}{4 over 5}{1 over 4}}$=${6 over 13}$.



Solution $2$: The required probability is $P(XYZ^c )={1 over 5}$.



So, my question is which of the solution is correct, and why the other one is incorrect, that is what is the question that the other solution makes that answer$?$






probability-theory




probability-theory






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asked Jan 4 at 12:30









Supriyo HalderSupriyo Halder

591113




591113












  • The probability that only C misses given that two out of three hit, is the same at the probability that only C misses.
    – Math-fun
    Jan 4 at 12:41


















  • The probability that only C misses given that two out of three hit, is the same at the probability that only C misses.
    – Math-fun
    Jan 4 at 12:41
















The probability that only C misses given that two out of three hit, is the same at the probability that only C misses.
– Math-fun
Jan 4 at 12:41




The probability that only C misses given that two out of three hit, is the same at the probability that only C misses.
– Math-fun
Jan 4 at 12:41










1 Answer
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Solution 1 is correct. The conditional probability (of being given that exactly two of the three participants hits the target) puts you in a restricted sample space. This is why you need to use the conditional probability formula of solution 1.






share|cite|improve this answer





















  • So what should be the question for the solution 2(to get solution 2 as the answer)?
    – Supriyo Halder
    yesterday










  • The question for which 'solution 2' is the answer would be "what is the probability that A and B both hit the target and C misses the target?".
    – paw88789
    yesterday










  • But, A and B both hit the target and C misses the target is same as exactly two of them hit the target and C missed the target.
    – Supriyo Halder
    14 hours ago










  • Yes, but in the actual problem, you are given the additional information that exactly two of the three hit the target. This additional information causes you to have a reduced sample space. That is we're interested in ABC' occurring but only out of the possibilities ABC', AB'C, A'BC. All the other possibilities are no longer relevant.
    – paw88789
    13 hours ago











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1 Answer
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1 Answer
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active

oldest

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active

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0














Solution 1 is correct. The conditional probability (of being given that exactly two of the three participants hits the target) puts you in a restricted sample space. This is why you need to use the conditional probability formula of solution 1.






share|cite|improve this answer





















  • So what should be the question for the solution 2(to get solution 2 as the answer)?
    – Supriyo Halder
    yesterday










  • The question for which 'solution 2' is the answer would be "what is the probability that A and B both hit the target and C misses the target?".
    – paw88789
    yesterday










  • But, A and B both hit the target and C misses the target is same as exactly two of them hit the target and C missed the target.
    – Supriyo Halder
    14 hours ago










  • Yes, but in the actual problem, you are given the additional information that exactly two of the three hit the target. This additional information causes you to have a reduced sample space. That is we're interested in ABC' occurring but only out of the possibilities ABC', AB'C, A'BC. All the other possibilities are no longer relevant.
    – paw88789
    13 hours ago
















0














Solution 1 is correct. The conditional probability (of being given that exactly two of the three participants hits the target) puts you in a restricted sample space. This is why you need to use the conditional probability formula of solution 1.






share|cite|improve this answer





















  • So what should be the question for the solution 2(to get solution 2 as the answer)?
    – Supriyo Halder
    yesterday










  • The question for which 'solution 2' is the answer would be "what is the probability that A and B both hit the target and C misses the target?".
    – paw88789
    yesterday










  • But, A and B both hit the target and C misses the target is same as exactly two of them hit the target and C missed the target.
    – Supriyo Halder
    14 hours ago










  • Yes, but in the actual problem, you are given the additional information that exactly two of the three hit the target. This additional information causes you to have a reduced sample space. That is we're interested in ABC' occurring but only out of the possibilities ABC', AB'C, A'BC. All the other possibilities are no longer relevant.
    – paw88789
    13 hours ago














0












0








0






Solution 1 is correct. The conditional probability (of being given that exactly two of the three participants hits the target) puts you in a restricted sample space. This is why you need to use the conditional probability formula of solution 1.






share|cite|improve this answer












Solution 1 is correct. The conditional probability (of being given that exactly two of the three participants hits the target) puts you in a restricted sample space. This is why you need to use the conditional probability formula of solution 1.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 4 at 12:48









paw88789paw88789

29k12349




29k12349












  • So what should be the question for the solution 2(to get solution 2 as the answer)?
    – Supriyo Halder
    yesterday










  • The question for which 'solution 2' is the answer would be "what is the probability that A and B both hit the target and C misses the target?".
    – paw88789
    yesterday










  • But, A and B both hit the target and C misses the target is same as exactly two of them hit the target and C missed the target.
    – Supriyo Halder
    14 hours ago










  • Yes, but in the actual problem, you are given the additional information that exactly two of the three hit the target. This additional information causes you to have a reduced sample space. That is we're interested in ABC' occurring but only out of the possibilities ABC', AB'C, A'BC. All the other possibilities are no longer relevant.
    – paw88789
    13 hours ago


















  • So what should be the question for the solution 2(to get solution 2 as the answer)?
    – Supriyo Halder
    yesterday










  • The question for which 'solution 2' is the answer would be "what is the probability that A and B both hit the target and C misses the target?".
    – paw88789
    yesterday










  • But, A and B both hit the target and C misses the target is same as exactly two of them hit the target and C missed the target.
    – Supriyo Halder
    14 hours ago










  • Yes, but in the actual problem, you are given the additional information that exactly two of the three hit the target. This additional information causes you to have a reduced sample space. That is we're interested in ABC' occurring but only out of the possibilities ABC', AB'C, A'BC. All the other possibilities are no longer relevant.
    – paw88789
    13 hours ago
















So what should be the question for the solution 2(to get solution 2 as the answer)?
– Supriyo Halder
yesterday




So what should be the question for the solution 2(to get solution 2 as the answer)?
– Supriyo Halder
yesterday












The question for which 'solution 2' is the answer would be "what is the probability that A and B both hit the target and C misses the target?".
– paw88789
yesterday




The question for which 'solution 2' is the answer would be "what is the probability that A and B both hit the target and C misses the target?".
– paw88789
yesterday












But, A and B both hit the target and C misses the target is same as exactly two of them hit the target and C missed the target.
– Supriyo Halder
14 hours ago




But, A and B both hit the target and C misses the target is same as exactly two of them hit the target and C missed the target.
– Supriyo Halder
14 hours ago












Yes, but in the actual problem, you are given the additional information that exactly two of the three hit the target. This additional information causes you to have a reduced sample space. That is we're interested in ABC' occurring but only out of the possibilities ABC', AB'C, A'BC. All the other possibilities are no longer relevant.
– paw88789
13 hours ago




Yes, but in the actual problem, you are given the additional information that exactly two of the three hit the target. This additional information causes you to have a reduced sample space. That is we're interested in ABC' occurring but only out of the possibilities ABC', AB'C, A'BC. All the other possibilities are no longer relevant.
– paw88789
13 hours ago


















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