The problem asks us to find the conditional probability that an animal will live beyond 25 years, given that it is currently 20 years old. We are given that the probability of living beyond 20 years from birth is 0.8, and the probability of living beyond 25 years from birth is 0.4. Let $A$ be the event that the animal lives beyond 20 years and $B$ be the event that the animal lives beyond 25 years. We want to find $P(B|A)$, which is the probability that the animal lives beyond 25 years given that it is already 20 years old.

1. Problem Description

The problem asks us to find the conditional probability that an animal will live beyond 25 years, given that it is currently 20 years old. We are given that the probability of living beyond 20 years from birth is 0.8, and the probability of living beyond 25 years from birth is 0.

4. Let $A$ be the event that the animal lives beyond 20 years and $B$ be the event that the animal lives beyond 25 years. We want to find $P(B|A)$, which is the probability that the animal lives beyond 25 years given that it is already 20 years old.

2. Solution Steps

We are given:

$P(A) = 0.8$ (probability of living beyond 20 years)

$P(B) = 0.4$ (probability of living beyond 25 years)

We want to find $P(B|A)$, the conditional probability of $B$ given $A$.

The formula for conditional probability is:

$P(B|A) = \frac{P(A \cap B)}{P(A)}$

If an animal lives beyond 25 years, it must also live beyond 20 years. Therefore, if event $B$ occurs, event $A$ must also occur. This means that the intersection of $A$ and $B$ is simply $B$, so $A \cap B = B$.

Thus, $P(A \cap B) = P(B) = 0.4$

Now we can calculate $P(B|A)$:

$P(B|A) = \frac{P(B)}{P(A)} = \frac{0.4}{0.8} = \frac{1}{2} = 0.5$

3. Final Answer

The probability that the animal will live beyond 25 years, given that it is currently 20 years old, is 0.5.