1. ## 20 questions

Imagine that you are taking a multiplechoice test. Four possible
answers are given for each question. You are to circle the correct
answer. Suppose there are 20 questions for which you don't know the
correct answer. You decide to guess for each of these questions. What
is the probability of anwsering a question correctly? How many of the 20 question can you get correct?

2. Originally Posted by gigi
Imagine that you are taking a multiplechoice test. Four possible
answers are given for each question. You are to circle the correct
answer. Suppose there are 20 questions for which you don't know the
correct answer. You decide to guess for each of these questions. What
is the probability of anwsering a question correctly? How many of the 20 question can you get correct?
The probability of answering a single question correctly is 0.25, because
choosing at random between the options gives 1 chance in four of being
correct.

The probability of getting at least one of the 20 correctly is 1 minus the
probability of getting them all wrong:

$
\rm {P} (n>0)=1-(1-p)^{20} \approx 0.9968
$

where $p$ is the probability of getting a single question right which
is as we have seen 0.25.

The probability of getting exactly one right is the sum of the probabilities
that: you get the first question right and the rest wrong, you get the second
question right and the rest wrong, .. ,you get the twentieth question right
and the rest wrong. But these are all the same and equal to: $p\times (1-p)^{19}$, so:

$
\rm{P(exactly\ 1\ right)}=20 \times p\times (1-p)^{19}\approx 0.021141
$

How many could you get right, well you could get all 20 right, but how many
should you expect to get right? That is the probability of getting a single
question right times the number of questions. The probability of getting a
single question right is 0.25, which means that you can expect to get a
quarter of all questions you guess on right, and a quarter of 20 is 5.

RonL