Problems Using Both Addition and Multiplicative Laws
Data Science and A.I. Lecture Series
PostNetwork Academy
Probability Laws
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- The addition law of probability states:
\[ P(A \cup B) = P(A) + P(B) – P(A \cap B) \]
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- The multiplicative law of probability for independent events states:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Example: Probability of Husband and Wife Selection
Problem:
- A husband and wife appear in an interview for two vacancies.
- Probability of husband’s selection: \( \frac{2}{5} \)
- Probability of wife’s selection: \( \frac{1}{5} \)
Solution
Let \( A \) be the husband’s selection and \( B \) be the wife’s selection.
Probability that both are selected:
\[ P(A \cap B) = P(A) \cdot P(B) = \frac{2}{5} \cdot \frac{1}{5} = \frac{2}{25} \]
Probability that at least one is selected:
\[ P(A \cup B) = P(A) + P(B) – P(A \cap B) \]
\[ P(A \cup B) = \frac{2}{5} + \frac{1}{5} – \frac{2}{25} = \frac{13}{25} \]
Example: Probability of Contradicting Statements
Problem:
- Person \( X \) speaks the truth in 80% of cases.
- Person \( Y \) speaks the truth in 90% of cases.
- Find the probability that they contradict each other.
Solution
Contradiction occurs when one speaks the truth while the other lies.
\[ P(T_X \cap F_Y) = 0.8 \times 0.1 = 0.08 \]
\[ P(F_X \cap T_Y) = 0.2 \times 0.9 = 0.18 \]
Total probability of contradiction:
\[ P(T_X \cap F_Y) + P(F_X \cap T_Y) = 0.08 + 0.18 = 0.26 \]
Video
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