To solve this problem, we can use the binomial probability formula. The probability of success (both parties in the labor force) is \( p = 0.521 \), and the probability of failure (not both parties in the labor force) is \( q = 1 - p = 0.479 \).
We are considering \( n = 5 \) couples, and we want to find the probability that more than three couples will have both persons in the labor force. This means we are looking for \( P(X > 3) \), where \( X \) is the number of couples with both persons in the labor force.
To find this, we can calculate:
\[ P(X > 3) = P(X = 4) + P(X = 5) \]
We can use the binomial probability formula:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]
Calculating \( P(X = 4) \):
\[ P(X = 4) = \binom{5}{4} (0.521)^4 (0.479)^{1} \]
First, calculate \( \binom{5}{4} = 5 \).
Next, compute \( (0.521)^4 \) and \( (0.479)^1 \):
\[ (0.521)^4 \approx 0.073529 \] \[ (0.479)^{1} \approx 0.479 \]
Now substitute these values into the formula:
\[ P(X = 4) = 5 \times 0.073529 \times 0.479 \approx 0.175428 \]
Calculating \( P(X = 5) \):
\[ P(X = 5) = \binom{5}{5} (0.521)^5 (0.479)^{0} \]
Here, \( \binom{5}{5} = 1 \), \( (0.479)^{0} = 1 \), and we compute \( (0.521)^5 \):
\[ (0.521)^5 \approx 0.038291 \]
Putting everything together:
\[ P(X = 5) = 1 \times 0.038291 \times 1 \approx 0.038291 \]
Adding Both Probabilities:
Now we combine the probabilities:
\[ P(X > 3) = P(X = 4) + P(X = 5) \approx 0.175428 + 0.038291 \approx 0.213719 \]
Rounding:
Rounding to the nearest thousandth, we get:
\[ P(X > 3) \approx 0.214 \]
Final Answer:
The probability that more than three of the couples will have both persons in the labor force is \( \boxed{0.214} \).
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