Calculating Probabilities: A Random Card Draw from 107 to 1006

In this article, we will walk through the process of determining the probability that a card drawn from a bag containing cards numbered from 107 to 1006 is not divisible by both 11 and 37. This involves a detailed breakdown of several mathematical concepts, including arithmetic sequences and the principle of inclusion-exclusion. Let's dive in.

Determining the Total Number of Cards

First, we need to calculate the total number of cards in the bag. The range of numbers is from 107 to 1006, inclusive. To find the total number of cards:

Total cards 1006 - 107 1 900

Identifying Numbers Divisible by 11

To find the cards divisible by 11, we need to determine the smallest and largest numbers in our range that are divisible by 11.

Smallest Number Divisible by 11

The smallest number divisible by 11 in the range can be found by taking the ceiling of 107 divided by 11 and multiplying by 11:

Smallest number lceil 107/11 rceil * 11 10 * 11 110

Largest Number Divisible by 11

The largest number divisible by 11 in the range can be found by taking the floor of 1006 divided by 11 and multiplying by 11:

Largest number lfloor 1006/11 rfloor * 11 91 * 11 1001

The sequence of numbers divisible by 11 from 110 to 1001 can be expressed using an arithmetic sequence formula:

110, 121, 132, ..., 1001

Using the arithmetic sequence formula to find the number of terms:

n (1001 - 110) / 11 1 882 / 11 1 82

Identifying Numbers Divisible by 37

Similarly, we identify the smallest and largest numbers in the range that are divisible by 37.

Smallest Number Divisible by 37

The smallest number divisible by 37 in the range is:

Smallest number lceil 107/37 rceil * 37 3 * 37 111

Largest Number Divisible by 37

The largest number divisible by 37 in the range is:

Largest number lfloor 1006/37 rfloor * 37 27 * 37 999

The sequence of numbers divisible by 37 from 111 to 999 can be expressed using an arithmetic sequence formula:

111, 148, 185, ..., 999

Using the arithmetic sequence formula to find the number of terms:

m (999 - 111) / 37 1 888 / 37 1 25

Identifying Numbers Divisible by Both 11 and 37 (Least Common Multiple)

The least common multiple of 11 and 37 is 407. We need to find the smallest and largest numbers in the range that are divisible by 407.

Smallest Number Divisible by 407

The smallest number divisible by 407 in the range is:

Smallest number lceil 107/407 rceil * 407 1 * 407 407

Largest Number Divisible by 407

The largest number divisible by 407 in the range is:

Largest number lfloor 1006/407 rfloor * 407 2 * 407 814

The sequence of numbers divisible by 407 from 407 to 814 is:

407, 814

Counting the number of terms:

2 terms

Applying the Principle of Inclusion-Exclusion

To find the total number of cards divisible by either 11 or 37, we use the principle of inclusion-exclusion:

Total cards divisible by 11 or 37 (Cards divisible by 11) (Cards divisible by 37) - (Cards divisible by 407)

Total cards divisible by 11 or 37 82 25 - 2 105

Calculating the Number of Cards Not Divisible by Either 11 or 37

The number of cards not divisible by 11 or 37 is given by:

Not divisible by 11 or 37 Total cards - (Divisible by 11 or 37) 900 - 105 795

Calculating the Probability

The probability that a randomly drawn card is not divisible by 11 or 37 is:

P(not divisible by 11 or 37) Not divisible by 11 or 37 / Total cards 795 / 900 89 / 100 0.89

Thus, the probability that the number on the card is not divisible by 11 and 37 is 89 / 100 or 0.89.