Assume a binomial probability distribution has p = .60 and n = 200.
a. What is the probability of 100 to 110 successes (to 4 decimals)?
b. What is the probability of 130 or more successes (to 4 decimals)?
To calculate the probabilities in a binomial distribution, we can use the formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes
- C(n, k) is the binomial coefficient, which is the number of ways to choose k 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
- n is the number of trials
a. To calculate the probability of 100 to 110 successes, we need to sum up the probabilities of getting exactly 100, 101, 102, ..., 110 successes. We can do this by using the formula mentioned above for each value and then summing up the results.
P(100 ≤ X ≤ 110) = P(X = 100) + P(X = 101) + P(X = 102) + ... + P(X = 110)
Substituting the values into the formula:
P(100 ≤ X ≤ 110) = [C(200, 100) * 0.60^100 * 0.40^100] + [C(200, 101) * 0.60^101 * 0.40^99] + ... + [C(200, 110) * 0.60^110 * 0.40^90]
Since calculating these probabilities manually can be time-consuming, we can use statistical software, online binomial calculators, or cumulative binomial probability tables to find the solution. For example, you can use a spreadsheet software like Excel or Google Sheets to set up the calculations and get the probabilities to 4 decimals.
b. To calculate the probability of 130 or more successes, we need to sum up the probabilities of getting 130, 131, 132, and so on up to the maximum possible value. Alternatively, we can calculate the complementary event, which is the probability of getting fewer than 130 successes, and subtract it from 1.
P(X ≥ 130) = 1 - P(X < 130)
P(X < 130) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 129)
Using the formula mentioned earlier, substitute the values and sum them up. Then, subtract the result from 1 to get P(X ≥ 130).
Similarly, you can use statistical software, online calculators, or cumulative binomial probability tables to find the solution.
Note: In this case, manual calculations may be cumbersome due to the large number of terms involved.
To find the probability in a binomial distribution, we will use the formula:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
where:
P(X = k) is the probability of getting k successes
n is the number of trials
p is the probability of success in each trial
k is the number of desired successes
Step 1: Calculate the probability of 100 to 110 successes.
Let's find the individual probabilities for 100 to 110 successes using the formula.
P(X = 100) = (200 C 100) * 0.6^100 * (1 - 0.6)^(200 - 100)
P(X = 101) = (200 C 101) * 0.6^101 * (1 - 0.6)^(200 - 101)
P(X = 102) = (200 C 102) * 0.6^102 * (1 - 0.6)^(200 - 102)
...
...
...
P(X = 110) = (200 C 110) * 0.6^110 * (1 - 0.6)^(200 - 110)
Step 2: Sum up the individual probabilities.
Add up all the individual probabilities to get the total probability:
P(100 to 110) = P(X = 100) + P(X = 101) + P(X = 102) + ... + P(X = 110)
Step 3: Calculate the probability of 130 or more successes.
To find the probability of 130 or more successes, we need to calculate the probabilities for 130, 131, 132, and so on until the maximum number of successes (which is 200 in this case).
P(X = 130) = (200 C 130) * 0.6^130 * (1 - 0.6)^(200 - 130)
P(X = 131) = (200 C 131) * 0.6^131 * (1 - 0.6)^(200 - 131)
P(X = 132) = (200 C 132) * 0.6^132 * (1 - 0.6)^(200 - 132)
...
...
...
P(X = 200) = (200 C 200) * 0.6^200 * (1 - 0.6)^(200 - 200)
Step 4: Sum up the individual probabilities.
Add up all the individual probabilities to get the total probability of 130 or more successes:
P(130 or more) = P(X = 130) + P(X = 131) + P(X = 132) + ... + P(X = 200)
Now you can calculate the probabilities using the calculator or a spreadsheet program.