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In a class of 27 students, 15 play an instrument and 13 play a sport. There are 3 students who do not play an instrument or a sport. What is the probability that a student who plays an instrument also plays a sport?
Answer:

In a class of $27$ students, $15$ play an instrument and $13$ play a sport. There are $3$ students who do not play an instrument or a sport. What is the probability that a student who plays an instrument also plays a sport? $\newline$ Answer:

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Probability of independent and dependent events

Full solution

Q. In a class of

27

students,

15

play an instrument and

13

play a sport. There are

3

students who do not play an instrument or a sport. What is the probability that a student who plays an instrument also plays a sport?

\newline

Answer:

Calculate Total Students: First, let's determine the total number of students who play either an instrument or a sport, or both. We know that $3$ students do not play an instrument or a sport, so the number of students who play at least one is $27 - 3$ . Calculation: $27 - 3 = 24$
Find Students in Both: Next, we need to find out how many students are involved in both activities. We can use the principle of inclusion-exclusion for this. The formula is: $\newline$ Number of students in both activities = (Number who play an instrument) + (Number who play a sport) - (Total number playing at least one) $\newline$ Calculation: $15 + 13 - 24 = 4$
Calculate Probability: Now we know that $4$ students play both an instrument and a sport. To find the probability that a student who plays an instrument also plays a sport, we divide the number of students who do both by the total number of students who play an instrument. $\newline$ Calculation: Probability $= \frac{\text{Number of students in both activities}}{\text{Number who play an instrument}}$ $\newline$ Probability $= \frac{4}{15}$

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