A husband and wife discover that there is a 11​% probability of their passing on a hereditary disease to any of their children. If they plan to have three children, what is the probability of the event that at least one child will inherit the​ disease?

To solve the problem, we will calculate the probability that at least one child will inherit the disease. The easiest way to do this is to first find the probability that **none** of the children inherit the disease and subtract this value from 1.

### Step-by-Step Solution:

1. **Given Data:**
   - Probability that a child inherits the disease: \( p = 0.11 \)
   - Probability that a child does **not** inherit the disease: \( 1 - p = 1 - 0.11 = 0.89 \)
   - They plan to have 3 children.

2. **Finding the probability that none of the children inherit the disease:**
   The probability that none of the children inherit the disease is the product of the probability of each child not inheriting it. Since each child is independent of the others:
   \[
   P(\text{none inherit}) = 0.89 \times 0.89 \times 0.89 = 0.89^3
   \]
   Calculating \( 0.89^3 \):
   \[
   P(\text{none inherit}) \approx 0.70497
   \]

3. **Finding the probability that at least one child inherits the disease:**
   The probability that at least one child inherits the disease is the complement of the probability that none inherit it:
   \[
   P(\text{at least one inherits}) = 1 - P(\text{none inherit})
   \]
   \[
   P(\text{at least one inherits}) = 1 - 0.70497 = 0.29503
   \]

Thus, the probability that at least one child will inherit the disease is approximately **0.295** or **29.5%**.

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Do you need any further clarifications or details? Here are 5 questions to expand on the topic:

1. How does the probability change if the couple has four children instead of three?
2. What is the probability that exactly two children inherit the disease?
3. How would the solution change if the probability of inheritance was higher, say 20%?
4. Can you calculate the probability that all three children inherit the disease?
5. What is the expected number of children who will inherit the disease?

**Tip:** When dealing with complementary events (like "at least one"), it's often easier to first calculate the complement and subtract from 1.

A husband and wife discover that there is an 11% probability of their passing on a hereditary disease to any of their children. If they plan to have three children, what is the probability of the event that at least one child will inherit the disease?

Calculate the probability that at least one child out of three will inherit a hereditary disease, given an 11% chance of passing it on. Explore key concepts in probability such as complementary events and independent trials, suitable for high school students in Grades 10-12.

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