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<4), n=5, p=0.6

Solution Steps

We are given a binomial distribution with parameters \(n = 5\) (number of trials) and \(p = 0.6\) (probability of success). We need to find the probability \(P(X < 4)\).

The probability of obtaining exactly \(k\) successes in a binomial distribution is given by:

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

where \(\binom{n}{k}\) is the binomial coefficient.

To find \(P(X < 4)\), we need to calculate the sum of probabilities for \(X = 0\), \(X = 1\), \(X = 2\), and \(X = 3\).

\[ P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \]

\[ P(X = 0) = \binom{5}{0} (0.6)^0 (0.4)^5 = 1 \times 1 \times 0.01024 = 0.01024 \]

\[ P(X = 1) = \binom{5}{1} (0.6)^1 (0.4)^4 = 5 \times 0.6 \times 0.0256 = 0.0768 \]

\[ P(X = 2) = \binom{5}{2} (0.6)^2 (0.4)^3 = 10 \times 0.36 \times 0.064 = 0.2304 \]

\[ P(X = 3) = \binom{5}{3} (0.6)^3 (0.4)^2 = 10 \times 0.216 \times 0.16 = 0.3456 \]

Add the probabilities calculated in the previous step:

\[ P(X < 4) = 0.01024 + 0.0768 + 0.2304 + 0.3456 = 0.6630 \]

Final Answer