Which of the following statements are true? I. P( ) = 0. II. P(S) = 1, where S is the sample space. III. If A B, then P(A) ≤ P(B). IV. P(A) + P(B) ≤ 1 for any two events.

Question 30

MediumOriginal question

Which of the following statements are true?
I. $P(\varnothing) = 0$ .
II. $P(S) = 1$ , where $S$ is the sample space.
III. If $A \subset B$ , then $P(A) \le P(B)$ .
IV. $P(A) + P(B) \le 1$ for any two events.

All four: I, II, III and IVCorrect answerYour answerI and II onlyCorrect answerYour answerI, II and III onlyCorrect answerYour answerI, II and IV onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

I and II are axioms. III: $B$ contains every outcome of $A$ and possibly more.
IV is false: on a die, 'even' has probability $\frac{1}{2}$ and 'less than 5' has $\frac{2}{3}$ , which add to more than 1, since the events overlap.

Worth noting

IV holds only for mutually exclusive events.

Concept: The axioms of probability
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsThe rules every probability follows

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Ten questions from Probability, at your own pace.

Which of the following statements are true?I. P(∅)=0P(\varnothing) = 0P(∅)=0.II. P(S)=1P(S) = 1P(S)=1, where SSS is the sample space.III. If A⊂BA \subset BA⊂B, then P(A)≤P(B)P(A) \le P(B)P(A)≤P(B).IV. P(A)+P(B)≤1P(A) + P(B) \le 1P(A)+P(B)≤1 for any two events.

Explanation

Which of the following statements are true?
I. $P(\varnothing) = 0$ .
II. $P(S) = 1$ , where $S$ is the sample space.
III. If $A \subset B$ , then $P(A) \le P(B)$ .
IV. $P(A) + P(B) \le 1$ for any two events.