Task: From a pool of $20$ lottery tickets labeled with different numbers from $1$ to $20$, we will draw $2$ tickets simultaneously. What is the probability that both drawn tickets will have prime numbers?
Thomas calculated:
(1) In the pool, there are a total of $8$ tickets labeled with prime numbers.
(2) The probability that the first drawn ticket will have a prime number is $\frac{8}{20}$.
(3) If the first drawn ticket is labeled with a prime number, then the probability that the second drawn ticket will also have a prime number is $\frac{8}{20}$.
(4) Therefore, the probability that both drawn tickets will have prime numbers is $\frac{8}{20}\cdot\frac{8}{20} = 0.16$.
Thomas's solution is incorrect. Determine where he made a mistake in his reasoning.
The mistake is in step (1). In the interval $\left[1; 20\right]$, there are only $7$ prime numbers. The probability that both drawn tickets will have prime numbers is $\frac{7}{20}\cdot\frac{7}{20}\approx 0.1225$.
The mistake is in step (3). If the first drawn ticket is labeled with a prime number, then the probability that the second drawn ticket will also have a prime number is $\frac{7}{19}$. The resulting probability is $\frac{8}{20}\cdot\frac{7}{19} \approx 0.1474$.
The mistake is in step (4). The probability that both drawn tickets will have prime numbers is $\frac{8}{20} + \frac{8}{20} = 0.8$.
The mistake is in step (4). The probability that both drawn tickets will have prime numbers is $1 - \left(\frac{8}{20} \cdot \frac{8}{20}\right) = 0.84$.