Question

The percentage of couples where both parties age in the labor force is 52.1. Choose five couples at random. Find the probability that fewer than 2 of the couples have both parties working. Round your answer to the nearest thousandth.

Answer 1

To solve this problem, we can model the situation using a binomial distribution. We're looking at the probability of fewer than 2 couples having both partners in the labor force out of 5 couples.

### Parameters:
- **n** (number of trials) = 5 (the number of couples)
- **p** (probability of success) = 0.521 (the probability that both parties in the couple are in the labor force)

We are looking for \( P(X < 2) \), where \( X \) is the number of couples with both parties working. This is the same as:

\[
P(X < 2) = P(X = 0) + P(X = 1)
\]

### Step 1: Calculate \( P(X = k) \)
The probability mass function for a binomial distribution is given by:

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

1. **Calculate** \( P(X = 0) \):
\[
P(X = 0) = \binom{5}{0} (0.521)^0 (0.479)^5
\]

The binomial coefficient \( \binom{5}{0} = 1 \), thus:
\[
P(X = 0) = 1 \cdot 1 \cdot (0.479)^5 \approx (0.479)^5 \approx 0.04741
\]

2. **Calculate** \( P(X = 1) \):
\[
P(X = 1) = \binom{5}{1} (0.521)^1 (0.479)^4
\]

The binomial coefficient \( \binom{5}{1} = 5 \), thus:
\[
P(X = 1) = 5 \cdot (0.521) \cdot (0.479)^4 \approx 5 \cdot 0.521 \cdot 0.05555 \approx 0.14445
\]

### Step 2: Combine the probabilities
Now we can sum these two probabilities up:
\[
P(X < 2) = P(X = 0) + P(X = 1) \approx 0.04741 + 0.14445 \approx 0.19186
\]

### Final Step: Round the answer
Thus, the probability that fewer than 2 of the couples have both parties working is:
\[
\boxed{0.192}
\]