In a class of 28 students, 16 play an instrument and 11 play a sport. There are 7 students who do not play an instrument or a sport. What is the probability that a student plays an instrument given that they play a sport?
The total number of students is 28.
There are 16 of them playing an instrument and 11 of them playing a sport. 7 students do not play either instrument or a sport.
The number of students playing is 28 – 7 = 21
The number of students playing both games is 16 + 11 – 21 = 6
The number of students playing sport only is 11 – 6 = 5
The number of students playing instrument only is 16 - 6 = 10
Now, the number of students playing sport or instrument = 10 + 5 = 15
Let "A" and "B" be the events that a student plays instrument and a sport respectively then,
"p(A)={16\\over28}\\\\p(B)={11\\over28}"
The probability we need to determine is,
"p(A|B)={p(A\\cap B)\\over p(B)}"
We have that,
"p(A\\cap B)={6\\over28}\\space and \\space p(B)={11\\over28}"
Thus,
"p(A|B)={{6\\over28}\\over{11\\over28}}={6\\over11}"
The probability that a student plays an instrument given that they play a sport is "{6\\over11}"
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