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Math  /  Data & Statistics

Question204 studenti hanno riempito un questionario riguardo agli sport praticati. In totale ci sono 88 studenti che giocano a Football, 92 che giocano a Cricket and 85 che giocano a Tennis. Ci sono 29 studenti che giocano solo a Tennis e a Football. Ci sono 15 studenti che giocano solo a Football e a Cricket. Ci sono 21 studenti che giocano solo a Tennis e a Cricket. Ci sono 4 studenti che giocano a tutti e tre gli sports. a. Quanti studenti praticano esattamente uno sport? □\square b. Quanti più studenti ci sono che non giocano a nessuno di questi sports rispetto al numero di studenti che giocano a tutti e tre? □\square c. Qual'è la probabilità che uno studente scelto a caso non giochi a nessuno di questi tre sports? Dai una risposta nella forma di una frazione ridotta ai minimi termini. □\square Finish attempt ...

Studdy Solution

STEP 1

1. The total number of students is 204.
2. The number of students playing each sport includes those who may play multiple sports.
3. We need to find the number of students who play exactly one sport, the number of students who play none of the sports, and the probability of selecting a student who plays none of the sports.

STEP 2

1. Use the principle of inclusion-exclusion to find the number of students playing exactly one sport.
2. Calculate the number of students who play none of the sports.
3. Determine the probability of selecting a student who plays none of the sports.

STEP 3

Let F F , C C , and T T represent the sets of students who play Football, Cricket, and Tennis, respectively.
Given: ∣F∣=88,∣C∣=92,∣T∣=85 |F| = 88, \quad |C| = 92, \quad |T| = 85 ∣F∩T∣=29,∣F∩C∣=15,∣T∩C∣=21 |F \cap T| = 29, \quad |F \cap C| = 15, \quad |T \cap C| = 21 ∣F∩C∩T∣=4 |F \cap C \cap T| = 4

STEP 4

Using the principle of inclusion-exclusion, calculate the number of students who play at least one sport:
∣F∪C∪T∣=∣F∣+∣C∣+∣T∣−∣F∩C∣−∣C∩T∣−∣T∩F∣+∣F∩C∩T∣|F \cup C \cup T| = |F| + |C| + |T| - |F \cap C| - |C \cap T| - |T \cap F| + |F \cap C \cap T|
=88+92+85−15−21−29+4=204= 88 + 92 + 85 - 15 - 21 - 29 + 4 = 204

STEP 5

Calculate the number of students who play exactly one sport:
∣F only∣=∣F∣−∣F∩C∣−∣F∩T∣+∣F∩C∩T∣|F \text{ only}| = |F| - |F \cap C| - |F \cap T| + |F \cap C \cap T| =88−15−29+4=48= 88 - 15 - 29 + 4 = 48
∣C only∣=∣C∣−∣C∩F∣−∣C∩T∣+∣F∩C∩T∣|C \text{ only}| = |C| - |C \cap F| - |C \cap T| + |F \cap C \cap T| =92−15−21+4=60= 92 - 15 - 21 + 4 = 60
∣T only∣=∣T∣−∣T∩F∣−∣T∩C∣+∣F∩C∩T∣|T \text{ only}| = |T| - |T \cap F| - |T \cap C| + |F \cap C \cap T| =85−29−21+4=39= 85 - 29 - 21 + 4 = 39
Total playing exactly one sport=∣F only∣+∣C only∣+∣T only∣\text{Total playing exactly one sport} = |F \text{ only}| + |C \text{ only}| + |T \text{ only}| =48+60+39=147= 48 + 60 + 39 = 147

STEP 6

Calculate the number of students who play none of the sports:
Students playing none=Total students−∣F∪C∪T∣\text{Students playing none} = \text{Total students} - |F \cup C \cup T| =204−204=0= 204 - 204 = 0

STEP 7

Calculate the probability that a randomly selected student plays none of the sports:
Probability=Number of students playing noneTotal students\text{Probability} = \frac{\text{Number of students playing none}}{\text{Total students}} =0204=0= \frac{0}{204} = 0
Solution: a. Students practicing exactly one sport: 147 \boxed{147} b. More students playing none than all three: 0 \boxed{0} c. Probability of selecting a student who plays none: 0 \boxed{0}

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