Let A and B be two events, the probability that at least one of them occurs is `4/5` and the probability that A and B occur simultaneously is `1/4`, then P(A') + P(B') is equal to ____

Answer

To solve the problem step by step, we will use the given probabilities and the properties of probability.

### Step 1: Understand the given information
We are given:
- \( P(A \cup B) = \frac{4}{5} \) (the probability that at least one of the events A or B occurs)
- \( P(A \cap B) = \frac{1}{4} \) (the probability that both events A and B occur simultaneously)

### Step 2: Use the formula for the union of two events
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values into the formula:
\[ \frac{4}{5} = P(A) + P(B) - \frac{1}{4} \]

### Step 3: Rearrange the equation to find \( P(A) + P(B) \)
To isolate \( P(A) + P(B) \), we can rearrange the equation:
\[ P(A) + P(B) = P(A \cup B) + P(A \cap B) \]
Substituting the known values:
\[ P(A) + P(B) = \frac{4}{5} + \frac{1}{4} \]

### Step 4: Find a common denominator and calculate
The common denominator for \( \frac{4}{5} \) and \( \frac{1}{4} \) is 20. Converting both fractions:
\[ \frac{4}{5} = \frac{16}{20}, \quad \frac{1}{4} = \frac{5}{20} \]
Now, add the two fractions:
\[ P(A) + P(B) = \frac{16}{20} + \frac{5}{20} = \frac{21}{20} \]

### Step 5: Calculate \( P(A') + P(B') \)
Using the complement rule, we know:
\[ P(A') = 1 - P(A), \quad P(B') = 1 - P(B) \]
Thus:
\[ P(A') + P(B') = (1 - P(A)) + (1 - P(B)) = 2 - (P(A) + P(B)) \]
Substituting the value we found:
\[ P(A') + P(B') = 2 - \frac{21}{20} \]
To calculate this, convert 2 to a fraction with a denominator of 20:
\[ 2 = \frac{40}{20} \]
Now perform the subtraction:
\[ P(A') + P(B') = \frac{40}{20} - \frac{21}{20} = \frac{19}{20} \]

### Final Answer
Thus, \( P(A') + P(B') = \frac{19}{20} \).

---