Calculating the Probability: Drawing a Multiple of 7 or 8 in a Lottery

Lottery draws can often be a source of curiosity, especially when we explore the statistical aspects involved. In this particular scenario, we are drawing six balls from a pool of 49 balls (numbered from 1 to 49). Let's dive into the probability of drawing a ball that is a multiple of 7 or 8.

Identifying Multiples

First, we identify the multiples of 7 and 8 within the range of 1 to 49:

Multiples of 7:

There are a total of 7 multiples of 7.

Multiples of 8:

There are a total of 6 multiples of 8.

Intersection of Multiples

Upon inspection, there are no common multiples of 7 and 8 within the range from 1 to 49. This means that there is no overlap between these sets.

Calculating the Probability

Now, let’s calculate the probability of drawing a ball that is a multiple of 7 or 8. Since there are no common multiples, we can simply add the number of favorable outcomes for multiples of 7 and 8.

Number of Favorable Outcomes:

Total multiples of 7: 7

Total multiples of 8: 6

Total favorable outcomes: 7 6 13

The total number of balls in the draw is 49.

Probability Calculation:

The probability of drawing a ball that is a multiple of 7 or 8 is given by:

( P frac{13}{49} )

Expressing the Probability in Decimal Form:

( P 0.265306 )

Or, in percentage form:

( P approx 26.53% )

Conclusion

Understanding the basic principles of probability and counting can help us clearly determine the likelihood of an event occurring. In this case, the probability of drawing a ball that is a multiple of 7 or 8 in a lottery draw consisting of 49 numbers is approximately 26.53%.

By breaking down the problem into smaller, manageable parts—identifying multiples and then calculating the union of those sets—we can efficiently solve such probability problems. This method is applicable to a variety of similar scenarios, making it a valuable skill to develop.