If E and F are events such that `P(E)=1/4, P(F)=1/2` and `P(E"and"F)=1/8`, find (i) `P(E"or"F)` (ii) `P` (not E and not F).

Explanation

To solve the problem, we will follow the steps outlined in the video transcript to find the required probabilities.

### Given:
- \( P(E) = \frac{1}{4} \)
- \( P(F) = \frac{1}{2} \)
- \( P(E \text{ and } F) = P(E \cap F) = \frac{1}{8} \)

### (i) Find \( P(E \text{ or } F) \)

To find \( P(E \text{ or } F) \), we use the formula:

\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]

Substituting the given values into the formula:

\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]
\[ = \frac{1}{4} + \frac{1}{2} - \frac{1}{8} \]

Next, we need a common denominator to add these fractions. The least common multiple of 4, 2, and 8 is 8.

Converting each term to have a denominator of 8:

\[ P(E) = \frac{1}{4} = \frac{2}{8} \]
\[ P(F) = \frac{1}{2} = \frac{4}{8} \]
\[ P(E \cap F) = \frac{1}{8} \]

Now substituting these values back into the equation:

\[ P(E \cup F) = \frac{2}{8} + \frac{4}{8} - \frac{1}{8} \]
\[ = \frac{2 + 4 - 1}{8} = \frac{5}{8} \]

Thus,

\[ P(E \text{ or } F) = \frac{5}{8} \]

### (ii) Find \( P(\text{not } E \text{ and not } F) \)

To find \( P(\text{not } E \text{ and not } F) \), we can use De Morgan's law, which states:

\[ P(\text{not } E \text{ and not } F) = P(\text{not } (E \text{ or } F)) = 1 - P(E \cup F) \]

We already calculated \( P(E \cup F) = \frac{5}{8} \). Now we substitute this into the equation:

\[ P(\text{not } E \text{ and not } F) = 1 - P(E \cup F) \]
\[ = 1 - \frac{5}{8} \]
\[ = \frac{8}{8} - \frac{5}{8} = \frac{3}{8} \]

Thus,

\[ P(\text{not } E \text{ and not } F) = \frac{3}{8} \]

### Final Answers:
(i) \( P(E \text{ or } F) = \frac{5}{8} \)

(ii) \( P(\text{not } E \text{ and not } F) = \frac{3}{8} \)

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