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A whole number from 1 to 15 inclusive is picked at random.
E = Whole numbers from 1 to 15 inclusive
M = Multiples of 3
F = Factors of 24
a) What is the probability that the number chosen is a multiple of 3 given that it is a factor of 24?
b) What is the probability that the number chosen is a factor of 24 given that it is a multiple of 3?
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Answer
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Step 1:Let's solve this step by step using set theory and probability.
Step 2:: Identify the sets
- F = Factors of 24: $$\{1, 2, 3, 4, 6, 8, 12, 24\}
- Total elements in E: |E| = 15
Step 3:: Find the intersection of M and F
- $$M \cap F = \{3, 6, 12\}
- Number of elements in M \cap F = 3
Step 4:: Solve part a) P(M | F)
- $$P(M | F) = \frac{|M \cap F|}{|F|} = \frac{3}{8}
- Probability of a multiple of 3 given it's a factor of 24
Step 5:: Solve part b) P(F | M)
- $$P(F | M) = \frac{|M \cap F|}{|M|} = \frac{3}{5}
- Probability of a factor of 24 given it's a multiple of 3
Final Answer
a) \frac{3}{8} b) \frac{3}{5}
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