If a seed is planted, it has a \( 60 \% \) chance of growing into a healthy plant. If 12 seeds are planted, what is the probability that exactly 1 doesn't grow?
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To find the probability that exactly 1 out of 12 seeds does not grow, we can model this scenario using the binomial probability formula. In this case, the probability of a seed growing is \( p = 0.6 \) and the probability of a seed not growing is \( q = 0.4 \). We want the probability of exactly \( k = 1 \) seed not growing out of \( n = 12 \) seeds planted: \[ P(X = k) = \binom{n}{k} p^{n-k} q^k \] Here, \( k = 1 \) (1 seed not growing), \( n = 12 \), \( p = 0.6 \), and \( q = 0.4 \). 1. First, calculate \( \binom{12}{1} = 12 \). 2. Then find \( p^{n-k} = (0.6)^{11} \) and \( q^k = (0.4)^{1} \). Now, putting it all together: \[ P(X = 1) = \binom{12}{1} (0.6)^{11} (0.4)^{1} \] Calculating \( (0.6)^{11} \) gives approximately \( 0.0367 \) and \( (0.4)^{1} \) is \( 0.4 \). So: \[ P(X = 1) = 12 \times 0.0367 \times 0.4 \approx 0.176 \, (rounded \, to \, three \, decimal \, places) \] Thus, the probability that exactly 1 out of 12 seeds does not grow is approximately \( 0.176 \).
