Understanding the Probability of a Prime Number Among 30 Tickets
Imagine you have 30 tickets, each numbered from 1 to 30. If one ticket is selected at random, what is the probability that the number on the ticket is a prime number? This question is a classic example in statistical probability. Let's break it down step-by-step.
Step 1: Identify the Prime Numbers from 1 to 30
A prime number is defined as a non-one number that has no integer factors other than 1 and itself. Let's list the prime numbers between 1 and 30:
2 3 5 7 11 13 17 19 23 29By identifying these primes, we can proceed to calculate the probability.
Step 2: Count the Prime Numbers
Upon counting the primes listed above, we find that there are 10 prime numbers between 1 and 30.
Step 3: Calculate the Total Number of Tickets
The total number of tickets is 30, as we are working with the numbers 1 to 30.
Step 4: Calculate the Probability
Using the formula for probability, which is the ratio of the number of favorable outcomes to the total number of outcomes, we can calculate the probability of drawing a ticket that shows a prime number:
[ P(text{Prime number}) frac{text{Number of prime numbers}}{text{Total number of tickets}} frac{10}{30} frac{1}{3} ]
Therefore, the probability that the ticket shows a prime number is ( frac{1}{3} ).
Practical Perspective
From a practical standpoint, the probability of drawing a prime number from the tickets can be simplified as follows:
Prime numbers within the range of 1 to 30 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. There are 10 prime numbers out of 30. Thus, the probability is: ( frac{10}{30} frac{1}{3} approx 0.333 )Theoretical Perspective
When we talk about probability in a more theoretical sense, it’s interesting to note that the probability of a randomly chosen number being prime is theoretically derived as follows:
The expression for the probability of a prime number is: ( 1-frac{6}{pi^2} )
This result is related to the famous Basel problem, which was solved by Euler. The Riemann-Zeta function has special properties that allow us to derive this probability.
For further reading, you might want to check out this video by Mathologer and this Wikipedia article on prime numbers.
In conclusion, the probability of drawing a ticket that shows a prime number from the tickets numbered 1 to 30 is ( frac{1}{3} ) or approximately 33.3%.