6.18 Calculate each binomial probability:

a. Fewer than 4 successes in 12 trials with a 10 percent chance of success.
b. At least 3 successes in 7 trials with a 40 percent chance of success.
c. At most 9 successes in 14 trials with a 60 percent chance of success.
d. More than 10 successes in 16 trials with an 80 percent chance of success.

b:0.38726

To calculate binomial probabilities, you can use the binomial probability formula:

P(x) = C(n, x) * p^x * (1-p)^(n-x)

Where:
- P(x) is the probability of getting x successes in n trials
- C(n, x) is the combination formula (n choose x), which calculates the number of ways to choose x successes from n trials
- p is the probability of success in a single trial
- (1-p) is the probability of failure in a single trial
- x is the number of successes
- n is the number of trials

Now let's use this formula to calculate the binomial probabilities for each case:

a. Fewer than 4 successes in 12 trials with a 10 percent chance of success.
In this case, x can be 0, 1, 2, or 3.
- P(0) = C(12, 0) * 0.1^0 * 0.9^12
- P(1) = C(12, 1) * 0.1^1 * 0.9^11
- P(2) = C(12, 2) * 0.1^2 * 0.9^10
- P(3) = C(12, 3) * 0.1^3 * 0.9^9

b. At least 3 successes in 7 trials with a 40 percent chance of success.
In this case, x can be 3, 4, 5, 6, or 7.
- P(3) = C(7, 3) * 0.4^3 * 0.6^4
- P(4) = C(7, 4) * 0.4^4 * 0.6^3
- P(5) = C(7, 5) * 0.4^5 * 0.6^2
- P(6) = C(7, 6) * 0.4^6 * 0.6^1
- P(7) = C(7, 7) * 0.4^7 * 0.6^0

c. At most 9 successes in 14 trials with a 60 percent chance of success.
In this case, x can be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.
- P(0) = C(14, 0) * 0.6^0 * 0.4^14
- P(1) = C(14, 1) * 0.6^1 * 0.4^13
- P(2) = C(14, 2) * 0.6^2 * 0.4^12
- ...
- P(9) = C(14, 9) * 0.6^9 * 0.4^5

d. More than 10 successes in 16 trials with an 80 percent chance of success.
In this case, x can be 11, 12, 13, 14, 15, or 16.
- P(11) = C(16, 11) * 0.8^11 * 0.2^5
- P(12) = C(16, 12) * 0.8^12 * 0.2^4
- P(13) = C(16, 13) * 0.8^13 * 0.2^3
- P(14) = C(16, 14) * 0.8^14 * 0.2^2
- P(15) = C(16, 15) * 0.8^15 * 0.2^1
- P(16) = C(16, 16) * 0.8^16 * 0.2^0

You can use the combination formula to calculate C(n, x), which is equal to n! / (x! * (n-x)!), where "!" denotes the factorial operation.