Question

Let $E1,E2,E3$ be three arbitrary of a sample space $S$. Consider the following statements which of the following statements are correct.

• A. $P\left(\text{only one of them occurs}\right)=P\left(E1\text{'}E2E3+E1E2\text{'}E3+E1E2E3\text{'}\right)$
• B. $P\left(\text{none of them occurs}\right)=P\left(E1\text{'}+E2\text{'}+E3\text{'}\right)$
• C. $P\left(\text{at least one of them occurs}\right)=P\left(E1+E2+E3\right)$
• D. $P\left(\text{all the three occurs}\right)=P\left(E1+E2+E3\right)$

Answer 1

The correct option is C

$P\left(\text{at least one of them occurs}\right)=P\left(E1+E2+E3\right)$

Explanation for correct option:

Option (C): $P\left(\text{at least one of them occurs}\right)=P\left(E1+E2+E3\right)$

Let $E1,E2,E3$ be three arbitrary of a sample space $S$

The probability that at least one of them occurs is $P\left(E1\cup E2\cup E3\right)$

$P\left(E1\cup E2\cup E3\right)=P\left(E1+E2+E3\right)$

Therefore, the correct answer is Option (C).

Explanation for incorrect options:

(A): $P\left(\text{only one of them occurs}\right)=P\left(E1\text{'}E2E3+E1E2\text{'}E3+E1E2E3\text{'}\right)$

The probability that only one of them occurs is $=P\left(E1\right)P\left(E2\text{'}\right)P\left(E3\text{'}\right)+P\left(E1\text{'}\right)P\left(E2\text{'}\right)P\left(E3\right)+P\left(E1\text{'}\right)P\left(E2\right)P\left(E3\text{'}\right)$

Therefore, option (A) is incorrect.

(B): $P\left(\text{none of them occurs}\right)=P\left(E1\text{'}+E2\text{'}+E3\text{'}\right)$

The probability that none of them occurs is $=P\left(E1\text{'}\right)P\left(E2\text{'}\right)P\left(E2\text{'}\right)$

Therefore, option (B) is incorrect.

(D): $P\left(\text{all the three occurs}\right)=P\left(E1+E2+E3\right)$

The probability that all the three occurs is $=1-P\left(E1\text{'}\right)P\left(E2\text{'}\right)P\left(E2\text{'}\right)$

Therefore, option (D) is incorrect.

Hence, option (C) is the correct answer.