A test consists of 12 multiple choice questions with 5 choices for each question. As an experiment, you GUESS on each and every answer without even reading the questions. Probability of getting less than 4 questions correct on this test is:

prog(right) = 1/5

prob(wrong) = 4/5

prob of your stated event
is, getting none right --> C(12,0)(1/5)^0 (4/5)^12 = .0687
getting one right --> C(12,1) (1/5) (4/5)^11 = .20616
getting two right --> C12,2) (1/5)^2 (4/5)^10 = .2835
getting three right --> C(12,3) (1/5)^3 (4/5)^9 = .23622

sum of these = appr .7946

btw, since passing is probably getting 6 or more correct, we have to do just two more cases for the failing situation
C(12,4)(1/5)^4(4/5)^8 = .1329
C(12,5) (1/5)^5 (4/5)^7 = .05315

for a total of .9806

Or you would have less than 2% chance of passing by merely guessing

To find the probability of getting less than 4 questions correct on this test, we can use the binomial probability formula.

The binomial probability formula is given by:

P(X = k) = C(n, 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 (12 questions in this case), k is the number of successful outcomes (less than 4 questions correct), C(n, k) is the combination formula to choose k items from a set of n items, p is the probability of guessing correctly (1/5 since there are 5 choices for each question), and (1 - p) is the probability of guessing incorrectly.

To find the probability of getting less than 4 questions correct, we need to find the sum of probabilities for k = 0, 1, 2, and 3.

P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

Now, let's calculate each term individually:

P(X = 0) = C(12, 0) * (1/5)^0 * (4/5)^12
= 1 * 1 * (4/5)^12
≈ 0.015

P(X = 1) = C(12, 1) * (1/5)^1 * (4/5)^11
= 12 * (1/5) * (4/5)^11
≈ 0.092

P(X = 2) = C(12, 2) * (1/5)^2 * (4/5)^10
= 66 * (1/25) * (4/5)^10
≈ 0.236

P(X = 3) = C(12, 3) * (1/5)^3 * (4/5)^9
= 220 * (1/125) * (4/5)^9
≈ 0.283

Now, let's calculate the sum of these probabilities:

P(X < 4) ≈ P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
≈ 0.015 + 0.092 + 0.236 + 0.283
≈ 0.626

Therefore, the probability of getting less than 4 questions correct on this test is approximately 0.626, or 62.6%.