I do not understand the binomial theorem. One of my questions is "there are 5 mutiple choice questions with 4 possible answers each. What is the probability of getting more than 3, exactly 3, and less than 3 correct?"
Thanks for any help:)
Use the binomial theorem to get the possibility of 0,1,2,3,4 and 5 correct.
For zero correct, the probability is just (3/4)^5 = 0.2373. 3/4 is the probability of getting each one wrong.
For five correct, the probability is
(1/4)^5 = 0.0001
For one correct, (one success and 4 failures) the probability is
5!/[1!*4!]*(1/4)*(3/4)^4
= 5(.25)(.3164) = 0.3955
(That is where you need the binomial theorem)
For two correct, using the same theorem, the probability is
[5!/(3!*2!)](1/4)^2*(3/4)^3
= 10*(0.25)^2(0.4219)= 0.2637
For three correct, the probability is
5!/[2!*3!)](1/4)^3*(3/4)^2 = 0.0879
For four correct, the probability is
(5!/4!)(1/4)^4*(3/4)= 5*.0039*.3164 = 0.0195
Use these results to get the probabilities for >3 and <3 right.
Use binomial the porem to expand (x+y) 25
To solve this problem, we can use the binomial probability formula, which is derived from the binomial theorem. The binomial probability formula calculates the probability of getting a specific number of successes in a fixed number of independent Bernoulli trials.
First, let's understand the terms used in the problem:
- Number of trials (n): The number of questions (5 in this case).
- Number of successes (k): The number of correct answers.
- Probability of success (p): The probability of getting a correct answer (1/4 in this case).
- Probability of failure (q): The probability of getting an incorrect answer (1 - p, which is 3/4 in this case).
Now, we can use the binomial probability formula to solve the problem:
1. Probability of getting more than 3 correct answers:
To find this probability, we need to sum up the probabilities of getting 4 correct answers and 5 correct answers.
P(X > 3) = P(X = 4) + P(X = 5)
Using the formula, the probability of getting exactly k successes in n trials is given by:
P(X = k) = (n choose k) * p^k * q^(n-k)
So, for P(X > 3):
P(X > 3) = P(X = 4) + P(X = 5)
= (5 choose 4) * (1/4)^4 * (3/4)^(5-4) + (5 choose 5) * (1/4)^5 * (3/4)^(5-5)
2. Probability of getting exactly 3 correct answers:
To find this probability, we use the binomial probability formula directly for P(X = 3).
P(X = 3) = (5 choose 3) * (1/4)^3 * (3/4)^(5-3)
3. Probability of getting less than 3 correct answers:
To find this probability, we need to sum up the probabilities of getting 0, 1, and 2 correct answers.
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
Using the formula, the probability of getting exactly 0, 1, and 2 successes will be calculated in a similar way.
With these formulas, you can easily calculate the probabilities.