To determine the probability that at least 2 of Hannah's 6 hens will lay eggs on a given day, we need to follow the steps below:
1. Understand the problem:
- We are given the probabilities of 0, 1, 2, 3, 4, and 5 hens laying eggs.
- We need to find the probability of either 2, 3, 4, 5, or all 6 hens laying eggs on a given day.
- But instead of directly summing these probabilities, we can use the fact that the total probability must add up to 1.
2. Sum the probabilities for 0 and 1 hens laying eggs:
- The probability that 0 hens lay eggs, [tex]\( P(X = 0) \)[/tex], is 0.000064.
- The probability that 1 hen lays an egg, [tex]\( P(X = 1) \)[/tex], is 0.002.
3. Calculate the probability of fewer than 2 hens laying eggs:
- Add the probabilities for 0 and 1 hens laying eggs:
[tex]\[
P(X < 2) = P(X = 0) + P(X = 1) = 0.000064 + 0.002 = 0.002064
\][/tex]
4. Calculate the probability of at least 2 hens laying eggs:
- Subtract the probability of fewer than 2 hens laying eggs from 1:
[tex]\[
P(X \geq 2) = 1 - P(X < 2) = 1 - 0.002064 = 0.997936
\][/tex]
Thus, the probability that at least 2 of Hannah’s hens will lay eggs on a given day is [tex]\( \boxed{0.998} \)[/tex].
