Questions: Assume the random variable X has a binomial distribution with the given probability of obtaining a success. Find the following probability, given the number of trials and the probability of obtaining a success. Round your answer to four decimal places. P(X ≤ 2), n=6, p=0.3

Solution Steps

Using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

For \( n = 6 \), \( x = 0 \), \( p = 0.3 \), and \( q = 1 - p = 0.7 \):

\[ P(X = 0) = \binom{6}{0} \cdot (0.3)^0 \cdot (0.7)^6 = 1 \cdot 1 \cdot 0.117649 = 0.1176 \]

Using the same formula for \( x = 1 \):

\[ P(X = 1) = \binom{6}{1} \cdot (0.3)^1 \cdot (0.7)^5 = 6 \cdot 0.3 \cdot 0.16807 = 0.3025 \]

Now for \( x = 2 \):

\[ P(X = 2) = \binom{6}{2} \cdot (0.3)^2 \cdot (0.7)^4 = 15 \cdot 0.09 \cdot 0.2401 = 0.3241 \]

To find \( P(X \leq 2) \), we sum the probabilities calculated:

\[ P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.1176 + 0.3025 + 0.3241 = 0.7442 \]

Final Answer