To find the probability that exactly 3 out of 6 seeds do not grow, we can use the binomial probability formula. In this scenario, the number of seeds that grow follows a binomial distribution.
Let:
- \( n = 6 \) (the number of seeds planted),
- \( p = 0.7 \) (the probability that a seed grows),
- \( q = 1 - p = 0.3 \) (the probability that a seed does not grow),
- \( k = 3 \) (the number of seeds that do not grow).
We want to find the probability that exactly 3 seeds do not grow, which means that 3 seeds do grow. The binomial probability formula is given by:
\[ P(X = k) = \binom{n}{k} p^{n-k} q^k \]
In this case, since we want the probability that exactly 3 do not grow, we can express it as:
\[ P(\text{exactly 3 do not grow}) = P(X = 3) = \binom{6}{3} (0.7)^{6-3} (0.3)^3 \]
Calculating step-by-step:
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Calculate \( \binom{6}{3} \): \[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
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Calculate \( (0.7)^{3} \): \[ (0.7)^3 = 0.343 \]
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Calculate \( (0.3)^{3} \): \[ (0.3)^3 = 0.027 \]
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Now combine all parts to find \( P(X = 3) \): \[ P(X = 3) = 20 \times 0.343 \times 0.027 \] \[ = 20 \times 0.009261 = 0.18522 \]
Thus, the probability that exactly 3 out of the 6 seeds do not grow is approximately 0.185 or 18.5%.
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