To find the probability [tex]\( P(x = 0) \)[/tex] in a Binomial distribution with parameters [tex]\( p = 0.4 \)[/tex] and [tex]\( n = 6 \)[/tex], we use the binomial probability formula:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
Given:
- [tex]\( p = 0.4 \)[/tex]: the probability of success in a single trial.
- [tex]\( n = 6 \)[/tex]: the number of trials.
- [tex]\( k = 0 \)[/tex]: the number of successes.
We need to calculate [tex]\( P(X = 0) \)[/tex]:
[tex]\[ P(X = 0) = \binom{6}{0} (0.4)^0 (0.6)^6 \][/tex]
First, calculate the binomial coefficient:
[tex]\[ \binom{6}{0} = \frac{6!}{0!(6-0)!} = 1 \][/tex]
Then, calculate the probability terms:
[tex]\[ (0.4)^0 = 1 \][/tex]
[tex]\[ (0.6)^6 = 0.046656 \][/tex]
Therefore:
[tex]\[ P(X = 0) = 1 \times 1 \times 0.046656 = 0.046656 \][/tex]
The calculated probability [tex]\( P(X = 0) = 0.046656 \)[/tex].
Among the given answer choices, the correct probability value calculated is not exactly listed. However, upon reviewing the closest probability value to [tex]\( 0.046656 \)[/tex] from the given options, the choice A: 0.062 seems like the most likely intended answer.
So, the correct choice to select is:
A. 0.062
