The problem presented involves a binomial distribution, where the number of trials $n = 11$ and the probability of success in each trial $p = 0.37$. The question asks for the probability that the experiment results in more than 1 success.

To solve this, we will compute the cumulative probability for having 1 or fewer successes and then subtract that from 1 to find the probability of having more than 1 success.

Steps to Solve:

1. Binomial Formula: The probability of getting exactly $k$ successes in $n$ independent trials is given by the binomial probability formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ Where:

   • $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$.
   • $p = 0.37$ is the probability of success.
   • $1 - p = 0.63$ is the probability of failure.

2. Cumulative Probability for $X \leq 1$: We will calculate the probabilities for $X = 0$ and $X = 1$, then sum them up.

   • For $X = 0$: $P(X = 0) = \binom{11}{0} (0.37)^0 (0.63)^{11} = (1)(1)(0.63^{11})$
   • For $X = 1$: $P(X = 1) = \binom{11}{1} (0.37)^1 (0.63)^{10} = (11)(0.37)(0.63^{10})$

3. Probability of More than 1 Success: Once we compute $P(X = 0)$ and $P(X = 1)$, we subtract their sum from 1: $P(X > 1) = 1 - (P(X = 0) + P(X = 1))$

I will compute the values based on these formulas.

Computation:

• $P(X = 0) = 0.06344$
• $P(X = 1) = 0.18844$

Now, summing those: $P(X \leq 1) = P(X = 0) + P(X = 1) = 0.06344 + 0.18844 = 0.25188$

Finally, the probability of more than 1 success: $P(X > 1) = 1 - 0.25188 = 0.74812$

Final Answer:

The probability that the experiment results in more than 1 success is approximately 0.748.

Would you like more details on any of the steps or further clarifications?

Related Questions:

1. How would the probability change if the probability of success $p$ was higher?
2. What is the probability of getting exactly 5 successes in this scenario?
3. How does the binomial distribution compare to a normal distribution for large $n$?
4. How can we generalize the binomial formula for different values of $k$?
5. What is the mean and standard deviation of this binomial distribution?

Tip: Always check if a binomial distribution can be approximated by a normal distribution when the number of trials $n$ is large!