In a Iottery of 50 tickets numbered 1 to 50 , two tickets are drawn simultaneously . Find the probability that
none of the tickets drawn has prime number ,

Answer

To find the probability that none of the tickets drawn has a prime number, we will follow these steps:

### Step 1: Identify the prime numbers between 1 and 50.
The prime numbers between 1 and 50 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

**Total prime numbers = 15.**

### Step 2: Calculate the total number of tickets that are not prime.
Since there are 50 tickets in total and 15 of them are prime, the number of tickets that are not prime is:
\[ 50 - 15 = 35 \]

### Step 3: Calculate the total number of ways to draw 2 tickets from 50.
The total number of ways to choose 2 tickets from 50 can be calculated using the combination formula:
\[ \text{Total outcomes} = \binom{50}{2} = \frac{50 \times 49}{2} = 1225 \]

### Step 4: Calculate the number of ways to draw 2 tickets from the 35 non-prime tickets.
The number of ways to choose 2 tickets from the 35 non-prime tickets is:
\[ \text{Favorable outcomes} = \binom{35}{2} = \frac{35 \times 34}{2} = 595 \]

### Step 5: Calculate the probability that none of the tickets drawn has a prime number.
The probability is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{none is prime}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{595}{1225} \]

### Step 6: Simplify the probability.
To simplify \(\frac{595}{1225}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 5:
\[ \frac{595 \div 5}{1225 \div 5} = \frac{119}{245} \]

### Final Answer:
The probability that none of the tickets drawn has a prime number is:
\[ \frac{119}{245} \]

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