A 10-item multiple choice test with 4 options (A,B,C,D) is given to a class in English, what is the probability that the student’s answers are of the same letters
let's pick one of the letters, say the A
Prob(A) = 1/4
you want that to happen 10 times in a row
prob(10 A's) = (1/4)^10
but the same reasoning can be used for each of the 4 letters, so
prob(10 of the same letter) = 4(1/4)^10 = (1/4)^9
To find the probability that the student's answers are all of the same letter, we need to determine the total number of possible combinations and then calculate the number of combinations where all the answers are of the same letter.
Each question has 4 answer options (A, B, C, D), so there are 4 choices for each of the 10 questions. This gives us a total of 4^10 possible combinations.
To calculate the number of combinations where all the answers are of the same letter, there are 4 options (A, B, C, D) to choose from. Therefore, there are 4 possible combinations where all the answers are of the same letter.
Therefore, the probability that the student's answers are all of the same letter is:
Number of combinations where all answers are of the same letter / Total number of possible combinations = 4 / 4^10
Simplifying the expression:
4 / 4^10 = 1 / 4^9 = 1 / 262,144
So the probability that the student's answers are all of the same letter is 1 in 262,144.