Let E and F be two events for which the probability of atleast one of them occurs is 3/4. what is the probability that neither E nor F occurs?

The hint is to use Demorgan's laws

Demorgan's laws state that the probability of the complement of the union of two events is equal to the complement of the intersection of their complements.

Let's denote the event of neither E nor F occurring as G.

From Demorgan's laws, we have:

P(G) = 1 - P(E ∪ F)

We are given that the probability of at least one of E or F occurring is 3/4. We can rewrite this as:

P(E ∪ F) = 3/4

So, we can substitute this value into the equation for P(G):

P(G) = 1 - (3/4)

Simplifying, we have:

P(G) = 1/4

Therefore, the probability that neither E nor F occurs is 1/4.

To solve this problem, we can use De Morgan's laws, which state:

1. The complement of the union of two events is equal to the intersection of their complements.
2. The complement of the intersection of two events is equal to the union of their complements.

Let's denote the probability of event E occurring as P(E) and the probability of event F occurring as P(F). We want to find the probability that neither E nor F occurs, which can be represented as the complement of (E ∪ F).

The complement of (E ∪ F) can be calculated as 1 - P(E ∪ F).

Now, using De Morgan's laws, we can rewrite the probability of the union of two events as the intersection of their complements:

P(E ∪ F) = 1 - P(E' ∩ F'),

where E' represents the complement of event E, and F' represents the complement of event F.

The given information states that the probability of at least one of the events E or F occurring is 3/4, which can be expressed as:

P(E ∪ F) = 3/4.

Substituting this into the equation above, we get:

1 - P(E' ∩ F') = 3/4.

Now, we can solve for P(E' ∩ F'):

P(E' ∩ F') = 1 - 3/4.

Simplifying,

P(E' ∩ F') = 1/4.

Finally, the probability that neither E nor F occurs (complement of (E ∪ F)) is given by P(E' ∩ F'), which we just calculated as 1/4. Therefore, the probability that neither E nor F occurs is 1/4.