NIMCET 2021 — Mathematics PYQ

Solution

1. Given Data

• $P(E)=0.25$

• $P(F)=0.50$

• $P(E\cap F)=0.14$ (Simultaneous occurrence)

2. Find the Probability of $E$ or $F$ occurring

Using the Addition Theorem of Probability:

$P(E\cup F)=P(E)+P(F)-P(E\cap F)$

$P(E\cup F)=0.25+0.50-0.14$

$P(E\cup F)=0.75-0.14=0.61$

3. Find the Probability that neither $E$ nor $F$ occur

The event "neither $E$ nor $F$" is denoted as $P(E^c\cap F^c)$ or $P(\overline{E\cup F})$. According to De Morgan's Law and the property of complementary events:

$P(\text{Neither }E\text{ nor }F)=1-P(E\cup F)$

$P(\text{Neither }E\text{ nor }F)=1-0.61$

$P(\text{Neither }E\text{ nor }F)=0.39$

Final Answer

The probability that neither $E$ nor $F$ occur is:

$0.39$

The correct option is 1.