Cards numbered 11 to 4040 are put in a bag. A card is drawn at random. Find the probability that the number on the card is a multiple of 33 or 55.

To solve this problem, we first need to find out how many multiples of 33 and 55 are there between 11 and 4040.

Step 1: Find the multiples of 33 The first multiple of 33 greater than 11 is 33 itself. The largest multiple of 33 less than 4040 is 4029. To find the total number of multiples, we subtract the smallest multiple from the largest, divide by 33, and then add 1.

(4029 - 33)/33 + 1 = 121 multiples of 33

Step 2: Find the multiples of 55 The first multiple of 55 greater than 11 is 55 itself. The largest multiple of 55 less than 4040 is 4015. To find the total number of multiples, we subtract the smallest multiple from the largest, divide by 55, and then add 1.

(4015 - 55)/55 + 1 = 73 multiples of 55

Step 3: Find the multiples of both 33 and 55 Since 33 and 55 have a common multiple (which is 165), there will be numbers which are counted twice. To find these, we need to find the multiples of 165 (which is the least common multiple of 33 and 55) between 11 and 4040.

The first multiple of 165 greater than 11 is 165 itself. The largest multiple of 165 less than 4040 is 3960. To find the total number of multiples, we subtract the smallest multiple from the largest, divide by 165, and then add 1.

(3960 - 165)/165 + 1 = 24 multiples of 165

Step 4: Subtract the multiples of both from the total multiples of 33 and 55 We then subtract the multiples of both 33 and 55 from the total multiples of 33 and 55 to avoid double counting.

121 + 73 - 24 = 170

Step 5: Find the total number of cards The total number of cards is 4040 - 11 + 1 = 4030

Step 6: Calculate the probability The probability is the number of favorable outcomes divided by the total number of outcomes.

170/4030 = 0.04218

So, the probability that the number on the card is a multiple of 33 or 55 is approximately 0.04218 or 4.218%.