True False[1 Marks ]

Question 11 Mark

State True or False for the statements:
If A, B and C are three independent events such that $P(A) = P(B) = P(C) = p$, then P (At least two of A, B, C occur) $=3 p^2-2 p^3$.

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Question 21 Mark

State True or False for the statements:
Let P(A) > 0 and P(B) > 0. Then A and B can be both mutually exclusive and independent.

Answer

False.

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Question 31 Mark

State True or False for the statements:
If A and B are independent, then.
P (exactly one of A, B occurs) = P(A)P(B') + P(B)P(A')

Answer

True.
Solution:
Exactly one of A and B occurs.
This meance if A occurs B does not occur and if B occurs A does not occur.
$\therefore$ Required probability $=\text{P}(\text{A}\cap\text{B}')+\text{P}(\text{A}'\cap\text{B})$
$=\text{P}(\text{A})\text{P}(\text{B}')+\text{P}(\text{A}')\text{P}(\text{B})$

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Question 41 Mark

State True or False for the statements:
Two independent events are always mutually exclusive.

Answer

False.
Explanation:
No, mutually exclusive events (with non-zero probability) are always dependent. The definition of independence for events A and B is that P(A and B) ... However, in the case that A and B are mutually exclusive, then P(A and B) = 0.

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Question 51 Mark

State True or False for the statements:
If A and B′ are independent events, then $\text{P}(\text{A}'\cup\text{B})=1-\text{P}(\text{A})\text{P}(\text{B'}).$

Answer

True.
Solution:
$\text{P}(\text{A}'\cup\text{B})=\text{P}(\text{A}')+\text{P}(\text{B})-\text{P}(\text{A}'\cap\text{B})$
$=\text{P}(\text{A}')+\text{P}(\text{B})-\big[\text{P}(\text{B})-\text{P}(\text{A}'\cap\text{B})\big]$
$=1-\text{P}(\text{A})+\text{P}(\text{B})-\text{P}(\text{B})-\text{P}(\text{A})\text{P}(\text{B})$
$=1-\text{P}(\text{A})+\text{P}(\text{A})\text{P}(\text{B})$
$=1-\text{P}(\text{A})\big[(1-\text{P}(\text{B})\big]$
$=1-\text{P}(\text{A})\text{P}(\text{B}')$

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Question 61 Mark

State True or False for the statements:
If A and B are two events such that P(A) > 0 and P(A) + P(B) >1, then $\text{P}\Big(\frac{\text{B}}{\text{A}}\Big)\geq1-\frac{\text{P}(\text{B}')}{\text{P}(\text{A})}.$

Answer

False.Solution:
$\because\text{P}\Big(\frac{\text{B}}{\text{A}}\Big)=\frac{\text{P}(\text{A}\cap\text{B})}{\text{P}(\text{A})}$
$=\frac{\text{P}(\text{A})+\text{P}(\text{B})-\text{P}(\text{A}\cup\text{B})}{\text{P}(\text{A})}>\frac{1-\text{P}(\text{A}\cup\text{B})}{\text{P}(\text{A})}$

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Question 71 Mark

State True or False for the statements:
If A and B are two independent events then P(A and B) = P(A) × P(B).

Answer

True.

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Question 81 Mark

State True or False for the statements:
If A and B are mutually exclusive events, then they will be independent also.

Answer

False.
Explanation:
The definition of being mutually exclusive (disjoint) means that it is impossible for two events to occur together. Given two events, A and B, they are mutually exclusive if $\text{(A} \cap \text{B)} = 0$. If these two events are mutually exclusive, they cannot be independent

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Question 91 Mark

State True or False for the statements:
If A and B are independent events, then A' and B' are also independent.

Answer

True.Solution:
Given A and B are independent. $\therefore\text{P}(\text{A}\cap\text{B})=\text{P}(\text{A})\cdot\text{P}(\text{B})$ $\text{P}(\text{A}'\cap\text{B}')=\text{P}[(\text{A}\cup\text{B})']$ $=1-\text{P}(\text{A}\cup\text{B})$ $=1-\big[\text{P}(\text{A})+\text{P}(\text{B})-\text{P}(\text{A}\cap\text{B})\big]$ $=1-\text{P}(\text{A})-\text{P}(\text{B})-\text{P}(\text{A})\cdot\text{P}(\text{B})$ $=\big[(1-\text{P}(\text{A})\big]\big[(1-\text{P}(\text{B})\big]=\text{P}(\text{A}')\text{P}(\text{B}')$ Thus A and B are independent.

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Question 101 Mark

State True or False for the statements:
Another name for the mean of a probability distribution is expected value.

Answer

True.
Explanation:
The mean of a probability distribution is the long-run arithmetic average value of a random variable having that distribution. In this context, it is also known as the expected value.

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Maths True False[1 Marks ]

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