Question

Which of the following statements are true or false? Give reasons for your answers.
(i) If P(A)= 0.4, P(A∪B)=0.7 and A and B are independent sets, then P(B)= 0.55.
(ii) By Chebyschev's inequality, P{|x-μ|≥2σ} ≤ 0.2

Answer 1

Answer on Question #66710 – Math – Statistics and Probability

Question

Which of the following statements are true or false? Give reasons for your answers.

(i) If $P(A) = 0.4$, $P(A \cup B) = 0.7$ and $A$ and $B$ are independent sets, then $P(B) = 0.55$.

(ii) By Chebyshev's inequality, $P\{|x - \mu| \geq 2\sigma\} \leq 0.2$

Solution

(i)

From independence of A and B it follows that

$$P(A \cap B) = P(A)P(B).$$

By inclusion-exclusion principle and by previous statement

$$P(A \cup B) = P(A) + P(B) - P(A \cap B) = P(A) + P(B) - P(A)P(B).$$

So we have

$$P(B) = \frac{P(A \cup B) - P(A)}{1 - P(A)} = \frac{0.7 - 0.4}{1 - 0.4} = \frac{0.3}{0.6} = 0.5 \neq 0.55.$$

The statement in the question is false.

(ii)

By Chebyshev's inequality

$$P(|x - \mu| \geq 2\sigma) \leq \frac{\sigma^2}{4\sigma^2} = \frac{1}{4} = 0.25 > 0.2.$$

The statement in the question is false.

Answer:

(i) False.

(ii) False.

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