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:
7 14 21 28 35 42 49There are a total of 7 multiples of 7.
Multiples of 8:
8 16 24 32 40 48There 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.