The problem in the image involves a binomial distribution. The task is to find the probability of getting 1 or fewer successes in a binomial experiment where the probability of success on a single trial is $p = 0.37$ and the number of trials is $n = 10$.

This means we need to calculate the cumulative probability of getting 0 or 1 successes.

The binomial probability formula is:

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

Where:

• $P(X = k)$ is the probability of exactly $k$ successes,
• $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$,
• $p$ is the probability of success,
• $n$ is the number of trials, and
• $k$ is the number of successes (in this case, either 0 or 1).

We need to calculate:

1. $P(X = 0)$
2. $P(X = 1)$
3. Add the two probabilities to get $P(X \leq 1)$.

Let me calculate that for you.The probability of obtaining 1 or fewer successes in this binomial experiment is approximately $0.068$ (rounded to three decimal places).

Would you like further details on the calculations or have any additional questions?

Here are five related questions you might explore:

1. How does the probability change if the number of trials increases?
2. What is the probability of getting exactly 2 successes?
3. How would you calculate the expected number of successes in this experiment?
4. How does changing the probability of success ($p$) affect the overall outcome?
5. What is the variance of the number of successes in a binomial distribution?

Tip: Binomial probabilities can be quickly calculated using a calculator or software, which is especially useful when dealing with larger values of $n$ or $k$.