To solve this problem, we can use the binomial probability formula. The scenario given involves a fixed number of trials (5 couples), each with two possible outcomes (both parties working or not), and the probability of success (both parties working) for each trial.
First, let's define the parameters:
- \( n = 5 \) (the number of couples)
- \( p = 0.521 \) (the probability that both parties are in the labor force)
- \( q = 1 - p = 0.479 \) (the probability that at least one party is not in the labor force)
We want to find the probability that fewer than 2 couples have both parties working, which includes the probabilities of having 0 or 1 couple working.
The binomial probability formula is: \[ P(X = k) = \binom{n}{k} p^k q^{n-k} \] where \( \binom{n}{k} \) is the binomial coefficient.
1. Calculate \( P(X = 0) \): \[ P(X = 0) = \binom{5}{0} (0.521)^0 (0.479)^5 = 1 \cdot 1 \cdot (0.479)^5 = 0.479^5 \] Calculating \( 0.479^5 \): \[ 0.479^5 \approx 0.047 \]
2. Calculate \( P(X = 1) \): \[ P(X = 1) = \binom{5}{1} (0.521)^1 (0.479)^4 = 5 \cdot (0.521) \cdot (0.479)^4 \] Calculating \( 0.479^4 \): \[ 0.479^4 \approx 0.098 \] Then calculating \( P(X = 1) \): \[ P(X = 1) \approx 5 \cdot 0.521 \cdot 0.098 \approx 0.255 \]
3. Now sum the probabilities for \( P(X < 2) = P(X = 0) + P(X = 1) \): \[ P(X < 2) \approx 0.047 + 0.255 \approx 0.302 \]
Finally, rounding to the nearest thousandth: \[ \boxed{0.302} \]
Thus, the probability that fewer than 2 of the couples have both parties working is approximately 0.302.
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