$\frac{9}{20}$
The problem asks for the probability that a number drawn randomly from the set {1, 2, 3, ..., 20} is divisible by 3 or 5.
Calculating Probability of Divisible Numbers
First, determine the total number of possible outcomes.
- Total numbers (sample space): The bag contains balls numbered 1 to 20. So, the total number of possible outcomes is $N = 20$.
Next, identify the favorable outcomes, which are numbers divisible by 3 or 5.
Numbers Divisible by 3
- The numbers between 1 and 20 that are divisible by 3 are: {3, 6, 9, 12, 15, 18}.
- The count of numbers divisible by 3 is $n(A) = 6$.
Numbers Divisible by 5
- The numbers between 1 and 20 that are divisible by 5 are: {5, 10, 15, 20}.
- The count of numbers divisible by 5 is $n(B) = 4$.
Numbers Divisible by Both 3 and 5
Numbers divisible by both 3 and 5 are divisible by their least common multiple (LCM), which is 15.
- The number between 1 and 20 that is divisible by 15 is: {15}.
- The count of numbers divisible by both 3 and 5 is $n(A \cap B) = 1$.
Total Favorable Outcomes (Divisible by 3 or 5)
Use the principle of inclusion-exclusion to find the total number of outcomes divisible by 3 or 5:
- $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
- $n(A \cup B) = 6 + 4 - 1 = 9$.
- There are 9 favorable outcomes.
Final Probability Calculation
The probability is the ratio of favorable outcomes to the total possible outcomes.
- Probability $P(A \cup B) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$
- $P(A \cup B) = \frac{n(A \cup B)}{N} = \frac{9}{20}$.
Therefore, the probability that the number on the ball drawn is divisible by 3 or 5 is $\frac{9}{20}$.
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Important Questions from Probability
Three dice are thrown. What is the probability of getting a sum which is a perfect square?
Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in its unit place?
Two dice are thrown. What is the probability that difference of numbers on them is 2 or 3 ?
Suppose that there is a chance for a newly constructed building to collapse, whether the design is faulty or not. The chance that the design is faulty is 10%. The chance that the building collapses is 95% if the design is faulty, otherwise it is 45%. If it is seen that the building has collapsed, then what is the probability that it is due to faulty design?
What is the probability that all three boys sit together?
