$\frac{9}{20}$
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
Two dice are thrown simultaneously. What is the probability of getting the same number on both the dice?
The probability of being 53 Sundays in year 2020 is-
Three dice are thrown randomly. The probability of coming 3 in at least one die is
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