Calculating Probabilities: Multiples of 2 or 3 Among 1 to 30
In this article, we will explore the probability of drawing a card that is a multiple of 2 or a multiple of 3 from a bag containing all the numbers from 1 to 30. This problem involves fundamental concepts in probability and arithmetic sequences, providing a clear and practical example for understanding these principles.
Total Numbers from 1 to 30
The total numbers from 1 to 30 can be counted directly, resulting in 30 numbers.
Counting Multiples of 2 and 3
To find the probability that a number drawn from the bag is a multiple of 2 or a multiple of 3, we first count the multiples of 2 and 3 separately and then use the principle of inclusion-exclusion to avoid double-counting.
Multiples of 2
The multiples of 2 in the range from 1 to 30 form an arithmetic sequence with the first term a 2 and the last term l 30, with a common difference d 2. Using the formula for the n-th term of an arithmetic sequence:
n (30 - 2) / 2 1 15
Thus, there are 15 multiples of 2 in the range from 1 to 30.
Multiples of 3
The multiples of 3 in the range from 1 to 30 also form an arithmetic sequence with the first term a 3 and the last term l 30, with a common difference d 3. Using the same formula:
n (30 - 3) / 3 1 10
Thus, there are 10 multiples of 3 in the range from 1 to 30.
Multiples of Both 2 and 3 (i.e., Multiples of 6)
The multiples of 6 in the range from 1 to 30 form an arithmetic sequence with the first term a 6, the last term l 30, and a common difference d 6. Again, using the formula:
n (30 - 6) / 6 1 5
Thus, there are 5 multiples of 6 in the range from 1 to 30.
Applying the Principle of Inclusion-Exclusion
The principle of inclusion-exclusion is used to find the total number of multiples of 2 or 3 by adding the number of multiples of 2 and the number of multiples of 3, then subtracting the number of multiples of both 2 and 3 (to avoid double-counting).
Number of multiples of 2 or 3 15 10 - 5 20
Calculating the Probability
The probability that a number drawn from the bag is a multiple of 2 or a multiple of 3 is calculated by dividing the number of favorable outcomes (multiples of 2 or 3) by the total number of possible outcomes (numbers from 1 to 30).
P(multiple of 2 or 3) 20 / 30 2/3
Thus, the probability that the number drawn is a multiple of 2 or a multiple of 3 is 2/3.
Application in Ticket Drawing
In another scenario, 20 tickets are selected which are multiples of 23 or 4. One ticket can be drawn in 20 ways out of 30 possible tickets. The probability is calculated as:
Required probability 20C1 / 30C1 20 / 30 2/3
This method confirms the same probability of 2/3.
Summary of Cases
The ticket numbers that fulfill the criteria are: 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30. There are 20 such cases.
Conclusion
In conclusion, the probability of drawing a multiple of 2 or 3 from a bag of the numbers 1 to 30 is 2/3, as demonstrated through the application of arithmetic sequences and the principle of inclusion-exclusion in the detailed steps provided above.