To find the binomial probabilities for the given values, we can use the binomial probability formula or a statistical software/calculator. The binomial distribution is defined by:

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

Where:

• $n$ is the number of trials,
• $r$ is the number of successes,
• $p$ is the probability of success on a single trial,
• $\binom{n}{r}$ is the binomial coefficient.

Given:

• $n = 200$
• $p = 0.65$

Let's find each probability using technology.

1. $P(r = 120)$

This is the probability of getting exactly 120 successes out of 200 trials.

2. $P(r \leq 90)$

This is the cumulative probability of getting 90 or fewer successes out of 200 trials.

3. $P(r \geq 140)$

This is the cumulative probability of getting 140 or more successes out of 200 trials.

Let's calculate these probabilities.The binomial probabilities are as follows:

1. $P(r = 120) \approx 0.020$
2. $P(r \leq 90) \approx 0.000$
3. $P(r \geq 140) \approx 0.078$

Would you like more details on how these probabilities were calculated or have any other questions?

Here are some related questions you might consider:

1. How do we derive the binomial coefficient in the probability formula?
2. What does the shape of the binomial distribution look like for $n = 200$ and $p = 0.65$?
3. How would the probabilities change if $p$ were different, say $p = 0.5$?
4. How can we approximate binomial probabilities using the normal distribution?
5. What are some practical applications of binomial distribution in real-world scenarios?

Tip: The binomial distribution is useful when you have a fixed number of independent trials, each with the same probability of success.