The probability that an electric bulb will last for 150 days or more is 0.7 and that it will last at the most 160 days is 0.8. Find the probability that it will last between 150 to... The probability that an electric bulb will last for 150 days or more is 0.7 and that it will last at the most 160 days is 0.8. Find the probability that it will last between 150 to 160 days? Read more

Steps to Solve

1. Define the Events

Let A be the event that the bulb lasts for 150 days or more. Let B be the event that the bulb lasts for at most 160 days.

2. Write down the given probabilities

$P(A) = 0.7$ $P(B) = 0.8$

3. Understand what we need to find

We need to find the probability that the bulb lasts between 150 and 160 days. This is the intersection of events A and B, i.e., $P(A \cap B)$.

4. Use the given hint

The hint $0.7 + 0.8 - P(A \cap B) = 1$ is based on the fact that $P(A \cup B) = 1$. This is because the event A or B must happen. A bulb will either last 150 days or more, or at most 160 days. Think of it another way: The range of days a bulb can last is from 0 to infinity. The first event covers the probability from 150 to infinity, and the second covers the probability from 0 to 160. Together, these events cover all possible outcomes, i.e. the entire range from zero to infinity. Therefore, that probability must be 1, or 100%. So, the probability of A union B is $P(A \cup B) = 1$.

5. Apply the union formula and solve for the intersection

The formula for the union of two events is: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Since $P(A \cup B) = 1$, we have: $1 = 0.7 + 0.8 - P(A \cap B)$

Rearrange to solve for $P(A \cap B)$: $P(A \cap B) = 0.7 + 0.8 - 1$ $P(A \cap B) = 1.5 - 1$ $P(A \cap B) = 0.5$