: Use technology to find the following binomial probabilities. Round to the nearest thousandth.

Find the probabilities for the values of n
 and \(p\_ when the conditions for the binomial distribution are met.

n=200
, p=0.65

P(r=120)=
 Answer 1 Question 6
P(r≤90)=
 Answer 2 Question 6
P(r≥140)=
 Answer 3 Question 6


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.

Use technology to find the following binomial probabilities. Round to the nearest thousandth. n=200, p=0.65. P(r=120), P(r≤90), P(r≥140)

Explore how to calculate binomial probabilities for n=200 and p=0.65. Learn how to find the probability for P(r=120), the cumulative probability P(r≤90), and P(r≥140), with detailed explanations of the binomial distribution. Suitable for students in Grades 11-12.

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