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In a class of 30 students, 6 play an instrument and 15 play a sport. There are 13 students who do not play an instrument or a sport. What is the probability that a student who plays an instrument does not play a sport? Answer:

In a class of 3030 students, 66 play an instrument and 1515 play a sport. There are 1313 students who do not play an instrument or a sport. What is the probability that a student who plays an instrument does not play a sport?\newlineAnswer:

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Q. In a class of 3030 students, 66 play an instrument and 1515 play a sport. There are 1313 students who do not play an instrument or a sport. What is the probability that a student who plays an instrument does not play a sport?\newlineAnswer:
  1. Determine Number of Students: Let's first determine the number of students who play only an instrument. We know that there are 66 students who play an instrument and 1313 students who do not play an instrument or a sport. Since there are 3030 students in total, we can find the number of students who play both an instrument and a sport by subtracting the number of students who play only an instrument and those who do not play anything from the total number of students.\newlineTotal number of students = Number of students who play only an instrument + Number of students who play both an instrument and a sport + Number of students who do not play anything\newline30=6+(Number of students who play both an instrument and a sport)+1330 = 6 + (\text{Number of students who play both an instrument and a sport}) + 13
  2. Solve for Both Instrument and Sport: Now, let's solve for the number of students who play both an instrument and a sport.\newline30=6+(Number of students who play both an instrument and a sport)+1330 = 6 + (\text{Number of students who play both an instrument and a sport}) + 13\newline30=19+(Number of students who play both an instrument and a sport)30 = 19 + (\text{Number of students who play both an instrument and a sport})\newline(Number of students who play both an instrument and a sport)=30−19(\text{Number of students who play both an instrument and a sport}) = 30 - 19\newline(Number of students who play both an instrument and a sport)=11(\text{Number of students who play both an instrument and a sport}) = 11
  3. Find Students Playing Only Instrument: Next, we need to find the number of students who play only an instrument. This can be found by subtracting the number of students who play both an instrument and a sport from the total number of students who play an instrument.\newlineNumber of students who play only an instrument == Total number of students who play an instrument −- Number of students who play both an instrument and a sport\newlineNumber of students who play only an instrument =6−11= 6 - 11
  4. Correct Error: Let's correct the error from the previous step. We need to find the number of students who play only an instrument, which is the total number of students who play an instrument minus the number of students who play both an instrument and a sport.\newlineNumber of students who play only an instrument == Total number of students who play an instrument −- Number of students who play both an instrument and a sport\newlineNumber of students who play only an instrument =6−(= 6 - (Number of students who play both an instrument and a sport))\newlineSince we have not yet determined the number of students who play both an instrument and a sport, we cannot complete this calculation yet. We need to find this number first.
  5. Find Students Playing Both Instrument and Sport: Let's find the number of students who play both an instrument and a sport. We know that there are 1515 students who play a sport and 1313 students who do not play an instrument or a sport. Therefore, the number of students who play a sport and possibly an instrument is:\newlineNumber of students who play a sport and possibly an instrument = Total number of students - Number of students who do not play an instrument or a sport\newlineNumber of students who play a sport and possibly an instrument = 30−1330 - 13\newlineNumber of students who play a sport and possibly an instrument = 1717
  6. Find Students Playing Only Sport: Now, we can find the number of students who play both an instrument and a sport by subtracting the number of students who play only a sport from the number of students who play a sport and possibly an instrument.\newlineNumber of students who play both an instrument and a sport == Number of students who play a sport and possibly an instrument −- Number of students who play only a sport\newlineSince we know that 1515 students play a sport, we can assume that the remaining 22 students (from the 1717 calculated in the previous step) must be the ones who play both an instrument and a sport.\newlineNumber of students who play both an instrument and a sport =17−15= 17 - 15\newlineNumber of students who play both an instrument and a sport =2= 2
  7. Calculate Probability: Now that we have the number of students who play both an instrument and a sport, we can find the number of students who play only an instrument.\newlineNumber of students who play only an instrument == Total number of students who play an instrument −- Number of students who play both an instrument and a sport\newlineNumber of students who play only an instrument =6−2= 6 - 2\newlineNumber of students who play only an instrument =4= 4
  8. Calculate Probability: Now that we have the number of students who play both an instrument and a sport, we can find the number of students who play only an instrument.\newlineNumber of students who play only an instrument == Total number of students who play an instrument −- Number of students who play both an instrument and a sport\newlineNumber of students who play only an instrument =6−2= 6 - 2\newlineNumber of students who play only an instrument =4= 4Finally, we can calculate the probability that a student who plays an instrument does not play a sport. This is the number of students who play only an instrument divided by the total number of students who play an instrument.\newlineProbability == Number of students who play only an instrument / Total number of students who play an instrument\newlineProbability == 46\frac{4}{6}\newlineProbability == 23\frac{2}{3}

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