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A bag contains 100 identical marble st...

A bag contains 100 identical marble stones which are numbered form 1 to 100. If one stone is a drawn at random from the bag, find the probability that it bears (i) a number divisible by 4 or 5 ,(ii) a number divisible by 4 and 5.

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To solve the problem, we will break it down into two parts as specified in the question. ### Part (i): Find the probability that the number drawn is divisible by 4 or 5. 1. **Identify the sample space**: The total number of outcomes is 100 since there are 100 marble stones numbered from 1 to 100. 2. **Find numbers divisible by 4**: - The numbers divisible by 4 from 1 to 100 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100. - Count: There are 25 numbers divisible by 4. 3. **Find numbers divisible by 5**: - The numbers divisible by 5 from 1 to 100 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100. - Count: There are 20 numbers divisible by 5. 4. **Find numbers divisible by both 4 and 5 (i.e., 20)**: - The numbers divisible by both 4 and 5 from 1 to 100 are: 20, 40, 60, 80, 100. - Count: There are 5 numbers divisible by both. 5. **Use the principle of inclusion-exclusion**: - Total numbers divisible by 4 or 5 = (Numbers divisible by 4) + (Numbers divisible by 5) - (Numbers divisible by both) - Total = 25 + 20 - 5 = 40. 6. **Calculate the probability**: - Probability (P) = (Number of favorable outcomes) / (Total outcomes) = 40 / 100 = 2 / 5. ### Part (ii): Find the probability that the number drawn is divisible by both 4 and 5. 1. **Identify the sample space**: The total number of outcomes is still 100. 2. **Find numbers divisible by both 4 and 5**: - As calculated earlier, the numbers divisible by both 4 and 5 are: 20, 40, 60, 80, 100. - Count: There are 5 numbers. 3. **Calculate the probability**: - Probability (P) = (Number of favorable outcomes) / (Total outcomes) = 5 / 100 = 1 / 20. ### Final Answers: - (i) The probability that the number drawn is divisible by 4 or 5 is **2/5**. - (ii) The probability that the number drawn is divisible by both 4 and 5 is **1/20**.
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