Suppose a multiple-choice test has 10 questions each with 5 choices. If a student guesses randomly. What is the probability that they get an B or better. ( they get at least a 80% on the exam) make sure to declare a binomial random variable and show all work
get 8 , 9 or 10 right
p right = .2
p not right = .8
binomial distribution
p 8 out of 10 = C(10,8)* .2^8 *.8^2
= 45 * 2.56*10^-6 * .64 = 73.7*10^-6
p 9 out of 10 = C(10,9)*.2^9 * .8^1
= 10 * 5.12*10^-7 * .8 = 4.096*10^-6
p 10 out of 10 = 1*.2^10 = .1024*10^-6
add 78 * 10^-6
78 out of a million, forget about it
To find the probability that a student guesses randomly and scores a B or better on the multiple-choice test, we can use the concept of binomial probability.
First, let's define a binomial random variable X, which represents the number of questions the student answers correctly:
X ~ B(n,p)
where n is the number of trials (number of questions) and p is the probability of success (probability of guessing correctly).
In this case, n = 10 (10 questions) and p = 1/5 (probability of guessing correctly since there are 5 choices for each question).
Now, let's calculate the probability of getting a B or better, which is equivalent to getting a score of 8, 9, or 10 on the exam.
P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10)
To calculate each probability, we can use the binomial probability formula:
P(X = k) = nCk * p^k * (1-p)^(n-k)
where nCk represents the binomial coefficient (n choose k).
Calculating each term:
P(X = 8) = 10C8 * (1/5)^8 * (4/5)^2
P(X = 9) = 10C9 * (1/5)^9 * (4/5)^1
P(X = 10) = 10C10 * (1/5)^10 * (4/5)^0
Using the combination formula nCk = n! / (k! * (n-k)!), where "!" denotes factorial:
10C8 = 10! / (8! * (10-8)!) = 45
10C9 = 10! / (9! * (10-9)!) = 10
10C10 = 10! / (10! * (10-10)!) = 1
Now we can calculate each probability:
P(X = 8) = 45 * (1/5)^8 * (4/5)^2
P(X = 9) = 10 * (1/5)^9 * (4/5)^1
P(X = 10) = 1 * (1/5)^10 * (4/5)^0
Next, we sum up these probabilities to get the probability of getting a B or better:
P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10)
Finally, calculate the probabilities:
P(X = 8) ≈ 0.000043
P(X = 9) ≈ 0.000003
P(X = 10) ≈ 0.00000002
P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10)
≈ 0.000043 + 0.000003 + 0.00000002
≈ 0.000046
Therefore, the probability that a student guessing randomly gets a B or better (scores at least an 80%) on the exam is approximately 0.000046 or 0.0046%.