In a survey of 30 students, it was found that 21 students  
played tennis, 21 played cricket and 18 played hockey. 
While 3 students played none of these sports, 18 played both 
tennis and cricket, 14 played both cricket and hockey and 
15 played both tennis and hockey. Find the probability that 
one randomly selected student from the group played all 
three sports.

We are given a set of students who play various sports, and we are tasked with finding the probability that a randomly selected student played all three sports (tennis, cricket, and hockey). Let’s break it down using set theory.

### Given dataWe are given a set of students who play various sports, and we are tasked with finding the probability that a randomly selected student played all three sports (tennis, cricket, and hockey). Let’s break it down using set theory.

### Given 
- Total number of students surveyed: \(30\)
- Students who played **tennis**: \(21\)
- Students who played **cricket**: \(21\)
- Students who played **hockey**: \(18\)
- Students who played **none** of these sports: \(3\)
- Students who played **both tennis and cricket**: \(18\)
- Students who played **both cricket and hockey**: \(14\)
- Students who played **both tennis and hockey**: \(15\)

Let:
- \(T\) = set of students who play tennis
- \(C\) = set of students who play cricket
- \(H\) = set of students who play hockey
- \(n(T \cap C \cap H)\) = number of students who play all three sports

We need to find \(n(T \cap C \cap H)\).

### Step 1: Calculate the number of students who played at least one sport
Since 3 students played none of these sports, the number of students who played at least one sport is:
\[
30 - 3 = 27
\]

### Step 2: Use the inclusion-exclusion principle
We use the inclusion-exclusion principle to count the number of students who played at least one sport. The principle is stated as:
\[
n(T \cup C \cup H) = n(T) + n(C) + n(H) - n(T \cap C) - n(C \cap H) - n(T \cap H) + n(T \cap C \cap H)
\]
Where:
- \(n(T) = 21\)
- \(n(C) = 21\)
- \(n(H) = 18\)
- \(n(T \cap C) = 18\)
- \(n(C \cap H) = 14\)
- \(n(T \cap H) = 15\)

We know that \(n(T \cup C \cup H) = 27\) (as calculated in step 1). Now substitute the known values into the inclusion-exclusion formula:
\[
27 = 21 + 21 + 18 - 18 - 14 - 15 + n(T \cap C \cap H)
\]
Simplify the equation:
\[
27 = 60 - 47 + n(T \cap C \cap H)
\]
\[
27 = 13 + n(T \cap C \cap H)
\]
Solve for \(n(T \cap C \cap H)\):
\[
n(T \cap C \cap H) = 27 - 13 = 14
\]

Thus, 14 students played all three sports.

### Step 3: Calculate the probability
The probability that a randomly selected student played all three sports is the ratio of the number of students who played all three sports to the total number of students:
\[
P(\text{all three sports}) = \frac{n(T \cap C \cap H)}{30} = \frac{14}{30} = \frac{7}{15}
\]

### Final Answer:
The probability that a randomly selected student played all three sports is \( \frac{7}{15} \).

---

Would you like more details or have any questions? 

Here are some related questions to expand on the concept:
1. What is the probability that a randomly selected student played only tennis?
2. How many students played exactly two sports?
3. What is the probability that a randomly selected student played at least two sports?
4. If we add one more sport, how would the inclusion-exclusion principle change?
5. How would the solution change if a student were allowed to play all sports except one?

**Tip:** The inclusion-exclusion principle is essential in solving problems involving overlapping sets, especially in probability and combinatorics.

In a survey of 30 students, it was found that 21 students played tennis, 21 played cricket and 18 played hockey. While 3 students played none of these sports, 18 played both tennis and cricket, 14 played both cricket and hockey and 15 played both tennis and hockey. Find the probability that one randomly selected student from the group played all three sports.

Learn how to calculate the probability that a randomly selected student played all three sports (tennis, cricket, and hockey) from a survey of 30 students. This solution uses set theory and the inclusion-exclusion principle to find that 14 students played all three sports, resulting in a probability of 7/15.

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