There is a 100 question multiple choice test each with 5 choices. You randomly guess each question.
Set up the binomial probability formula to find the probability of getting exactly 80 correct out of 100.
Check to see if you can use the normal approximation to binomial probabilities. then use normal approximation to compute the probability of getting 80 or more correct.
prob(right ) = 1/5
prob(wrong) = 4/5
let the number that are right be x
C(100,80) (1/5)^80 (4/5)^20
I am stymied for a method to evaluate this, my calculator cannot handle it
Wolfram says:
http://www.wolframalpha.com/input/?i=evaluate+C%28100%2C80%29+%281%2F5%29%5E80+%284%2F5%29%5E20+
I got that but i am not sure how to do the bottom part. when i use normal approximation to commute it i get 0%
notice the result is
7.47 x 10^-38 which I would consider close to zero.
To set up the binomial probability formula to find the probability of getting exactly 80 correct out of 100, we can use the following formula:
P(x=k) = (n C k) * p^k * (1-p)^(n-k)
where:
P(x=k) is the probability of getting exactly k correct answers,
n is the number of trials or questions (100 in this case),
k is the number of successful outcomes or correct answers (80 in this case),
p is the probability of getting a correct answer on a single trial (1/5 since there are 5 choices for each question),
n C k is the binomial coefficient or the number of possible combinations of k successes out of n trials.
Using this formula, the probability of getting exactly 80 correct out of 100 can be calculated.
Now, to check if we can use the normal approximation to binomial probabilities, we need to ensure that both np and n(1-p) are greater than 5. Here, n is 100 and p is 1/5.
np = 100 * (1/5) = 20
n(1-p) = 100 * (4/5) = 80
Since both np and n(1-p) are greater than 5, we can use the normal approximation.
To use the normal approximation, we can approximate the binomial distribution with a normal distribution. The mean of the normal distribution is np, and the standard deviation is √(np(1-p)).
In this case, the mean is 20 and the standard deviation is √(20 * (4/5)).
Once we have these values, we can calculate the probability of getting 80 or more correct using the normal distribution by calculating the z-score and finding the corresponding area under the curve.
The z-score for 80 can be calculated using the formula:
z = (x - mean) / standard deviation,
where x is the value we're interested in (80 in this case).
After calculating the z-score, we can use a standard normal distribution table or a calculator to find the area under the curve to the left of the z-score. Since we're looking for the probability of getting 80 or more correct, we then subtract this probability from 1 to get the final result.