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QuestionFind the numbers for athletes and musicians in a Venn diagram given P(A∣B)=715P(A|B) = \frac{7}{15}, with 25 athletes and 15 musicians.

Studdy Solution

STEP 1

Assumptions1. The total number of athletes offered admission is25 (Event A). . The total number of musicians offered admission is15 (Event B).
3. The probability that an athlete is also a musician is 715\frac{7}{15}.

STEP 2

First, we need to find the number of athletes who are also musicians. We can do this by multiplying the total number of athletes by the given probability.
Number of athletes who are also musicians=Total number of athletestimesProbability of an athlete being a musicianNumber\, of\, athletes\, who\, are\, also\, musicians = Total\, number\, of\, athletes \\times Probability\, of\, an\, athlete\, being\, a\, musician

STEP 3

Now, plug in the given values for the total number of athletes and the probability to calculate the number of athletes who are also musicians.
Number of athletes who are also musicians=25times715Number\, of\, athletes\, who\, are\, also\, musicians =25 \\times \frac{7}{15}

STEP 4

Calculate the number of athletes who are also musicians.
Number of athletes who are also musicians=25times715=11.67Number\, of\, athletes\, who\, are\, also\, musicians =25 \\times \frac{7}{15} =11.67

STEP 5

Since the number of students cannot be a fraction, we round it to the nearest whole number. In this case, we round down to11 because we assume that we cannot have a partial student.
Number of athletes who are also musicians=11Number\, of\, athletes\, who\, are\, also\, musicians =11

STEP 6

Now that we have the number of athletes who are also musicians, we can find the number of athletes who are not musicians and the number of musicians who are not athletes.For athletes who are not musiciansNumber of athletes who are not musicians=Total number of athletes−Number of athletes who are also musiciansNumber\, of\, athletes\, who\, are\, not\, musicians = Total\, number\, of\, athletes - Number\, of\, athletes\, who\, are\, also\, musiciansFor musicians who are not athletesNumber of musicians who are not athletes=Total number of musicians−Number of athletes who are also musiciansNumber\, of\, musicians\, who\, are\, not\, athletes = Total\, number\, of\, musicians - Number\, of\, athletes\, who\, are\, also\, musicians

STEP 7

Plug in the values for the total number of athletes, the total number of musicians, and the number of athletes who are also musicians to calculate the number of athletes who are not musicians and the number of musicians who are not athletes.
For athletes who are not musiciansNumber of athletes who are not musicians=25−11Number\, of\, athletes\, who\, are\, not\, musicians =25 -11For musicians who are not athletesNumber of musicians who are not athletes=15−11Number\, of\, musicians\, who\, are\, not\, athletes =15 -11

STEP 8

Calculate the number of athletes who are not musicians and the number of musicians who are not athletes.
For athletes who are not musiciansNumber of athletes who are not musicians=25−11=14Number\, of\, athletes\, who\, are\, not\, musicians =25 -11 =14For musicians who are not athletesNumber of musicians who are not athletes=15−11=4Number\, of\, musicians\, who\, are\, not\, athletes =15 -11 =4So, in the Venn diagram, the number of students in Event A only is14, in Event B only is4, and in both Events A and B is11.

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