Part One: 12 college students attempt the same multiple choice question once each. The multiple choice question has eight possible choices, with only one being correct. If all 12 college students make a random guess, what is the probability that at least one college student will guess the correct answer?
Part Two: The same conditions (as per Part One) are repeated across 60 independent multiple choice questions (i.e., events). Again, there are 12 college students completing each of the 60 different questions. Each of the 60 different questions have 8 possible choices, with only one being correct. What is the combined probability that at least one college student will guess the correct answer for each of the 60 multiple choice questions?
[NOTE: I am NOT asking what the probability of a single college student guessing 60 out of 60 is; I am suggesting that for each of the 60 events (or questions) all students have equal likelihood of guessing a correct answer. The students AND the events are independent. Basically, what is the aggregated probability solution from Part One across 60 independent events].
Much obliged :-)
Each college student has a probablity of 1/8. The probability all 12 students getting the answer wrong can be calculated by $(7/8)^{12}$ since each student has a 7/8 chance of getting it wrong and the events are independent. $1-(7/8)^{12}$ gives us the probability that at least one student gets the answer correct. For the second part we can calculate the probablity of one student getting all answers correct, which is $(1/8)^{60}$, so the probability of getting at least one wrong is $1-(1/8)^{60}$. The probablity of ALL 12 students getting at least one wrong is $(1-(1/8)^{60})^{12}$ The complement of the last probabilty is equivalent to at least one student getting all the correct answers, which is $1-(1-(1/8)^{60})^{12}$ and extremely close to 0.