Step 1: List the sets A and B
A = {10, 11, 12, 13, 14, 15, ..., 28, 29, 30}
B = {9, 18, 27, 36, 45}
Step 2: Identify the elements in A ∩ B (the intersection)
- Multiples of 9 less than 50 are: 9, 18, 27, 36, 45.
- Which of these are in A (10 to 30)? 18 and 27.
Thus A ∩ B = {18, 27}
Step 3: Compute sizes
- |A|: A contains integers from 10 through 30 inclusive. Number of elements = 30 - 10 + 1 = 21.
- |B|: 5 elements as listed above.
- |A ∩ B|: 2 elements (18 and 27).
Step 4: Probability formulas
Assuming a random element chosen from the larger universal set that contains both A and B is not specified, we typically treat probabilities conditioned on the event by listing outcomes within the conditioning set.
Interpretation options:
- If we treat the random experiment as choosing an element uniformly from B and asking P(A|B) = P(A ∩ B) / P(B) = |A ∩ B| / |B|.
- If we treat the random experiment as choosing an element uniformly from A and asking P(B|A) = P(A ∩ B) / P(A) = |A ∩ B| / |A|.
Step 5: Compute the probabilities
- P(A | B) = |A ∩ B| / |B| = 2 / 5 = 0.4
- P(B | A) = |A ∩ B| / |A| = 2 / 21 ≈ 0.0952
Step 6: Interpret the results
- Given B, the probability that the number is also in A is 0.4 (since 2 of the 5 numbers in B lie in A).
- Given A, the probability that the number is also in B is 2/21 ≈ 9.5% (since only 2 of the 21 numbers in A are in B).