A farmer plants 12 saplings. On average, 15% of saplings planted fail to survive their first winter. Find the probability that at least two of his saplings will die in that first winter.

To find the probability that at least two out of 12 saplings will die in the first winter, we need to use the binomial probability formula. 

Given:
- \( n = 12 \) (number of trials, i.e., the number of saplings planted)
- \( p = 0.15 \) (probability of failure for each sapling)

We define \( X \) as the random variable representing the number of saplings that die. \( X \) follows a binomial distribution \( X \sim \text{Binomial}(n, p) \).

We need to find \( P(X \geq 2) \), the probability that at least 2 saplings will die.

The binomial probability formula is:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

We can compute \( P(X \geq 2) \) as:
\[
P(X \geq 2) = 1 - P(X < 2) = 1 - (P(X = 0) + P(X = 1))
\]

Now, let's calculate \( P(X = 0) \) and \( P(X = 1) \):

\[
P(X = 0) = \binom{12}{0} (0.15)^0 (0.85)^{12} = 1 \cdot 1 \cdot (0.85)^{12} = (0.85)^{12}
\]

\[
P(X = 1) = \binom{12}{1} (0.15)^1 (0.85)^{11} = 12 \cdot 0.15 \cdot (0.85)^{11}
\]

Using a calculator, we find:

\[
(0.85)^{12} \approx 0.1425
\]

\[
12 \cdot 0.15 \cdot (0.85)^{11} \approx 12 \cdot 0.15 \cdot 0.1673 \approx 0.3011
\]

Now sum these probabilities:

\[
P(X < 2) = P(X = 0) + P(X = 1) \approx 0.1425 + 0.3011 = 0.4436
\]

Finally, we calculate \( P(X \geq 2) \):

\[
P(X \geq 2) = 1 - P(X < 2) = 1 - 0.4436 = 0.5564
\]

Thus, the probability that at least two of the saplings will die in the first winter is approximately \( 0.5564 \) or 55.64%.

Would you like further details or have any questions?

Here are 8 related questions for further exploration:
1. What is the probability that exactly three saplings will die?
2. What is the probability that none of the saplings will die?
3. How would the probability change if the survival rate was 90% instead?
4. What is the expected number of saplings that will survive?
5. How does the standard deviation of the number of saplings that die look like?
6. If the farmer plants 20 saplings, what is the new probability that at least two will die?
7. How would the probability change if the number of saplings increased to 50?
8. How can this binomial distribution be approximated using a normal distribution?

**Tip:** When dealing with binomial probabilities, ensure you understand the parameters \(n\) (number of trials) and \(p\) (probability of success) as they are crucial in determining the outcome.

Calculate the probability that at least two out of 12 saplings planted by a farmer will die in their first winter. This problem uses the binomial probability formula to determine the likelihood of sapling survival based on given failure rates. Explore detailed step-by-step solutions and related questions.

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