Suppose that 45% of all babies bom in a particular hospital are boys. If 8 bables born in the hospital are randomly selected, what is the probability that fewer than 3 of them are boys?

Carry your intermediate computations to at least four decimal places, and round your answer to two decimals places.

To solve this problem, we'll use the binomial distribution, which is appropriate here since we are dealing with a fixed number of independent trials (in this case, 8 babies) and each trial has only two possible outcomes (boy or girl).

### Step 1: Define the binomial distribution parameters
- **Number of trials (n):** 8 (since 8 babies are randomly selected)
- **Probability of success (p):** 0.45 (the probability that a baby is a boy)
- **Number of successes (k):** We need to find the probability that fewer than 3 babies are boys, so \( k \) can be 0, 1, or 2.

### Step 2: Binomial Probability Formula
The probability of getting exactly \( k \) boys out of \( n \) babies is given by the binomial probability formula:

\[
P(X = k) = \binom{n}{k} \times p^k \times (1 - p)^{n - k}
\]

where \( \binom{n}{k} \) is the binomial coefficient, calculated as:

\[
\binom{n}{k} = \frac{n!}{k! \times (n - k)!}
\]

### Step 3: Calculate the probabilities for \( k = 0, 1, 2 \)

1. **For \( k = 0 \):**
   \[
   P(X = 0) = \binom{8}{0} \times (0.45)^0 \times (0.55)^8
   \]
   \[
   P(X = 0) = 1 \times 1 \times (0.55)^8 = 0.0043
   \]

2. **For \( k = 1 \):**
   \[
   P(X = 1) = \binom{8}{1} \times (0.45)^1 \times (0.55)^7
   \]
   \[
   P(X = 1) = 8 \times 0.45 \times (0.55)^7 = 0.0266
   \]

3. **For \( k = 2 \):**
   \[
   P(X = 2) = \binom{8}{2} \times (0.45)^2 \times (0.55)^6
   \]
   \[
   P(X = 2) = 28 \times (0.2025) \times (0.55)^6 = 0.0913
   \]

### Step 4: Add the probabilities for \( k = 0, 1, 2 \)

\[
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
\]
\[
P(X < 3) = 0.0043 + 0.0266 + 0.0913 = 0.1222
\]

### Step 5: Round to two decimal places
The final probability, rounded to two decimal places, is:

\[
\boxed{0.12}
\]

Would you like more details on this or have any other questions? 

Here are some related questions to explore:

1. What is the probability of having exactly 3 boys out of 8 babies?
2. How does the probability change if the percentage of boys born increases to 50%?
3. What would the probability be if 10 babies were selected instead of 8?
4. How can we calculate the cumulative probability using a binomial distribution table?
5. What is the expected number of boys in 8 babies?
6. How would the variance and standard deviation of the number of boys be calculated?
7. What is the probability of having at least 5 boys out of 8 babies?
8. How would the probabilities change if only 5 babies were randomly selected instead of 8?

**Tip:** When dealing with binomial distribution, it's often useful to recognize patterns or use statistical software to handle more complex calculations, especially when dealing with larger values of \( n \).

Explore the probability of fewer than 3 boys out of 8 babies using the binomial distribution formula. This problem involves calculating probabilities based on the given percentage of boys born in a hospital. Learn step-by-step with detailed computations and solutions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation