Calculating Probability: Cards Divisible by 5 or 6

In this article, we delve into a problem that requires us to calculate the probability of drawing a card that is divisible by 5 or 6 from a set of cards numbered from 31 to 50. Understanding the principles behind probability and divisibility is essential in solving this problem.

Introduction

We start with a set of cards numbered from 31 to 50. Our goal is to find the probability that a randomly drawn card is exactly divisible by 5 or 6. This involves identifying the total number of cards, determining the count of cards divisible by 5 and 6, and applying the principle of inclusion-exclusion.

Step 1: Identify the Range of Numbers

The set of cards includes the numbers from 31 to 50. This gives us a total of 20 numbers:

31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50

Here, the total number of outcomes (cards) is 20.

Step 2: Find Numbers Divisible by 5

To identify the numbers divisible by 5, we look for numbers in the range 31 to 50 that are multiples of 5:

35, 40, 45, 50

There are 4 such numbers.

Step 3: Find Numbers Divisible by 6

Next, we identify the numbers in the range 31 to 50 that are divisible by 6:

36, 42, 48

There are 3 such numbers.

Step 4: Find Numbers Divisible by Both 5 and 6

The least common multiple (LCM) of 5 and 6 is 30. We check if there are any numbers in the range 31 to 50 that are divisible by 30:

There are no such numbers in our range.

Therefore, the count of numbers divisible by both 5 and 6 is 0.

Step 5: Use the Principle of Inclusion-Exclusion

To find the total number of favorable outcomes (numbers divisible by 5 or 6), we use the principle of inclusion-exclusion:

[ text{Total} text{Divisible by 5} text{Divisible by 6} - text{Divisible by Both 5 and 6} ]

Plugging in the values, we get:

[ text{Total} 4 3 - 0 7 ]

So, there are 7 favorable outcomes.

Step 6: Calculate the Probability

The probability ( P ) that a drawn card is divisible by 5 or 6 is:

[ P frac{text{Number of favorable outcomes}}{text{Total outcomes}} frac{7}{20} ]

Thus, the probability is 0.35 or 35%.

Conclusion

The probability that the drawn card is exactly divisible by 5 or 6 is 7 out of 20, or 0.35.