91Ó°ÊÓ

Let's organize the given information: Number of trials, n = 10 Probability of success, p = 0.4 Probability of failure, q = 1 - p = 0.6 We will use the binomial formula for probability \(P(X=k) =C(n,k) p^k q^{n-k}\) and the cumulative binomial formula (sum of probabilities).

To calculate \(P(x=4)\) we can use the binomial formula given as: $$ P(x=4) = C(10,4)\cdot 0.4^4 \cdot 0.6^6 $$ Now, calculating the values: $$ P(x=4) = \binom{10}{4} \cdot 0.4^4 \cdot 0.6^6 \approx 0.2508 $$

For these two probabilities, we will use the cumulative probability formula. We first calculate \(P(x < 4)\) and then subtract it from 1 to get the probability \(P(x \geq 4)\). Next, subtract the probability we computed in step 2 (\(P(x=4)\)) to get \(P(x>4)\). 1. \(P(x \geq 4) = 1 - P(x<4) = 1 - (P(x=0) + P(x=1) + P(x=2) + P(x=3))\) Calculate each probability individually: $$P(x=0) = \binom{10}{0} \cdot 0.4^0 \cdot 0.6^{10} \approx 0.0060$$ $$P(x=1) = \binom{10}{1} \cdot 0.4^1 \cdot 0.6^9 \approx 0.0401$$ $$P(x=2) = \binom{10}{2} \cdot 0.4^2 \cdot 0.6^8 \approx 0.1209$$ $$P(x=3) = \binom{10}{3} \cdot 0.4^3 \cdot 0.6^7 \approx 0.2150$$ Now sum these probabilities and subtract the result from 1. $$P(x \geq 4) = 1 - 0.0060 - 0.0401 - 0.1209 - 0.2150 \approx 0.6179$$ 2. \(P(x > 4) = P(x \geq 4) - P(x=4)\) $$P(x > 4) = 0.6179 - 0.2508 \approx 0.3671$$

Since we have already calculated the individual probabilities, we can just add them up. $$P(x \leq 4) = P(x=0) + P(x=1) + P(x=2) + P(x=3) + P(x=4) = 0.0060 + 0.0401 + 0.1209 + 0.2150 + 0.2508 \approx 0.6329$$

Now, we can calculate \(\mu\), the mean of a binomial random variable with n=10 trials and a probability p=0.4 of success using the formula \(\mu=np\). $$ \mu = 10 \cdot 0.4 = 4 $$

Finally, we can calculate the standard deviation of a binomial random variable with n=10 trials, a probability p=0.4 of success, and a probability q=0.6 of failure using the formula \(\sigma=\sqrt{npq}\). $$ \sigma = \sqrt{10 \cdot 0.4 \cdot 0.6} = \sqrt{2.4} \approx 1.549 $$ Now, we have our results: a. \(P(x=4) \approx 0.2508\) b. \(P(x \geq 4) \approx 0.6179\) c. \(P(x>4) \approx 0.3671\) d. \(P(x \leq 4) \approx 0.6329\) e. \(\mu = 4\) f. \(\sigma \approx 1.549\)