Understanding the Probability of Drawing Odd Tickets from a Box: A Comprehensive Guide
Imagine a box filled with tickets, each uniquely numbered from 1 to 25. If you were to randomly draw two tickets one after the other without replacement, what would the probability be that both tickets drawn are odd? This question not only tests basic probability concepts but also the understanding of independent and dependent events. In this article, we'll break down the problem step by step to find the solution, ensuring you have a clear grasp of the underlying principles.
Identifying the Total Number of Tickets
The box contains a total of 25 tickets, numbered 1 to 25. Each ticket is equally likely to be drawn from the box.
Counting the Odd-Numbered Tickets
Among these tickets, we need to identify the odd numbers. These are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, and 25. Counting these, we see there are 13 odd tickets in total.
Calculating the Probability of Drawing Two Odd Tickets
To find the probability that both drawn tickets will show odd numbers, we must calculate the combined probability of two successive events: the first ticket drawn is odd, and the second ticket drawn is also odd, given that the first one was odd.
First Draw (Probability of Drawing an Odd Ticket)
The probability of drawing an odd ticket on the first draw is the ratio of the number of odd tickets to the total number of tickets:
P1
Second Draw (Probability of Drawing an Odd Ticket Given the First One Was Odd)
After the first odd ticket has been drawn, there are now 12 odd tickets left out of a total of 24 tickets. Thus, the probability of drawing an odd ticket on the second draw is:
P2|1
Combined Probability
The combined probability of both events happening (drawing two odd tickets one after the other) is the product of the probabilities of each individual event:
Pboth odd P1 times; P2|1 times;
Exploring Additional Aspects
Let's further illustrate the solution using a different approach. Using the fundamental principle of counting, we calculate the total number of ways to choose 2 tickets out of 25, and then the number of ways to choose 2 odd tickets out of 13. This can help verify our previous calculation.
Total possible outcomes for drawing 2 tickets from 25:
25 choose 2 (
Outcomes where both tickets are odd:
13 choose 2 (
Therefore, the probability of both tickets being odd is:
Pboth odd ()
Independent Events and Probability
It's important to understand that the two draws are dependent events. However, the probability of the second event (drawing an odd number after the first one has been drawn) can be treated as an independent event given the condition that the first ticket drawn is odd.
Conclusion
In conclusion, the probability that both tickets drawn will show odd numbers is (). This example not only reinforces the principles of probability and independent events but also demonstrates the importance of considering the changing conditions after each event in a sequence of dependent events.