The probability of succeeding with a new business idea is 0.442 . If you have 10 business ideas what is the probability that 2 or fewer of the ideas will be successful? (Round your answer to 3 decimal places if necessary.) Provide your answer belont \[ P(X \leq 2)=\square \]
Solution
Step 1 ：Given that the probability of succeeding with a new business idea is \(p = 0.442\). If you have \(n = 10\) business ideas, we are asked to find the probability that 2 or fewer of the ideas will be successful.
Step 2 ：We can calculate this by finding the probabilities for 0, 1, and 2 successes and then summing these probabilities.
Step 3 ：The probability of 0 successes is calculated as \(\binom{n}{0} \cdot p^0 \cdot (1-p)^{n-0}\), which equals \(0.0029264559366789233\).
Step 4 ：The probability of 1 success is calculated as \(\binom{n}{1} \cdot p^1 \cdot (1-p)^{n-1}\), which equals \(0.023180887527098275\).
Step 5 ：The probability of 2 successes is calculated as \(\binom{n}{2} \cdot p^2 \cdot (1-p)^{n-2}\), which equals \(0.0826286474756245\).
Step 6 ：Summing these probabilities gives us the total probability of 2 or fewer successes, which equals \(0.1087359909394017\).
Step 7 ：Rounding this to 3 decimal places, we get the final answer: \(P(X \leq 2)=\boxed{0.109}\).
