Probability of Drawing a Ticket with a Multiple of 3 or 8: A Comprehensive Analysis

Imagine a scenario where a box contains a total of 320 tickets, each numbered from 1 to 320. This article delves into the calculation of the probability of drawing a ticket that has a multiple of either 3 or 8. Our analysis will provide a detailed breakdown of the steps and calculations involved in determining this probability, making it a valuable resource for students and professionals alike.

Understanding the Sample Space

First, we need to establish the sample space (S) of our experiment. The sample space comprises all possible outcomes. In this case, the sample space is defined as all numbers from 1 to 320:

S {1, 2, 3, ..., 320}

The number of outcomes in the sample space (nS) is:

nS 320

Defining the Events

We are interested in the event (E) of drawing a ticket that has a multiple of either 3 or 8. To calculate this, we first need to identify the individual events:

Event A: Multiple of 3

Event A represents the event that a number is a multiple of 3. To determine the number of favorable outcomes for this event, we can use the formula:

PA [320/3]/320

Here, [320/3] represents the integer part of 320 divided by 3, which is 106. Therefore:

PA 106/320

Event B: Multiple of 8

Event B represents the event that a number is a multiple of 8. Similarly, to determine the number of favorable outcomes for this event, we use the formula:

PB [320/8]/320

Here, [320/8] represents the integer part of 320 divided by 8, which is 40. Therefore:

PB 40/320

Intersection of Events (A and B)

Event A and B have some overlapping numbers, specifically those that are multiples of both 3 and 8, i.e., multiples of 24. To avoid double-counting these numbers, we need to determine the number of outcomes in the intersection of A and B:

PAB [320/24]/320

Here, [320/24] represents the integer part of 320 divided by 24, which is 13. Therefore:

PAB 13/320

Applying Inclusion-Exclusion Principle

To find the total number of favorable outcomes for event E (both A and B), we use the inclusion-exclusion principle:

nE nA nB - nA∩B

From our earlier calculations, we have:

nA 106

nB 40

nA∩B 13

Substituting these values into the formula:

nE 106 40 - 13 133

Calculating the Probability of Event E

The probability of event E (drawing a ticket that is a multiple of 3 or 8) is given by the ratio of the number of favorable outcomes to the total number of outcomes in the sample space:

PE nE / nS

Substituting the values we calculated:

PE 133 / 320

This simplifies to:

PE ≈ 0.416

Conclusion

In conclusion, the probability of drawing a ticket that is a multiple of 3 or 8 from a box containing 320 tickets numbered from 1 to 320 is approximately 0.416 or 41.6%. This analysis provides a clear and comprehensive understanding of the calculation involved, highlighting the importance of the inclusion-exclusion principle in dealing with overlapping events.

Understanding such probabilities is crucial in various fields, including statistics, probability theory, and real-world decision-making scenarios. By breaking down the problem into manageable parts and applying the correct mathematical principles, we can tackle more complex problems with confidence.

For more in-depth studies and further reading, we recommend exploring related topics such as probability theory and combinatorial mathematics.