6. What is the expected value of the binomial distributions where n = 16 and P=0.85?
a) 6
b) 7.4
c) 12.4
d)13.6

The expected value \( E(X) \) of a binomial distribution is given by the formula:

\[
E(X) = n \times P
\]

where \( n \) is the number of trials, and \( P \) is the probability of success on each trial.

Given:
- \( n = 16 \)
- \( P = 0.85 \)

Plugging these values into the formula:

\[
E(X) = 16 \times 0.85 = 13.6
\]

Therefore, the expected value is **13.6**, which corresponds to option:

**d) 13.6**

Would you like more details on this calculation, or do you have any questions?

Here are 5 related questions for further exploration:

1. What is the variance of this binomial distribution?
2. How do you calculate the standard deviation of a binomial distribution?
3. What is the probability of getting exactly 13 successes in this distribution?
4. How does changing the probability \( P \) affect the expected value of a binomial distribution?
5. What is the probability of getting at least 14 successes in this distribution?

**Tip:** Remember that for binomial distributions, the expected value indicates the average number of successes expected over a large number of trials.

Understand how to calculate the expected value of a binomial distribution with n = 16 and P = 0.85. Learn the formula E(X) = n * P and its application in probability theory. Suitable for high school students studying probability and statistics.

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