A test has five multiple-choice questions. Each question has 4 answer choices, of which 1 is correct answer and the other 3 are incorrect. Suppose that you guess all the answers. a) How many ways are there to answer the 5 questions? b) What is the probability of getting all 5 questions right? c) What is the probability of getting exactly 4 questions right and 1 wrong? d) What is the probability of doing well (getting at least 4 right)?

On the other hand, the probability of get one question right is [tex]\frac{1}{4}[/tex] or 0.25. Because there is only one answer correct from 4 possibles answers.

At the same way the probability of gt one question wrong is 3/4 or 0.75. Because there are 3 answers incorrect from 4 possible choices.

So, the probability of get 5 questions right, is the multiplication of probabilities of get the right answer in every question:

P= 0.25*0.25*0.25*0.25*0.25=0.000976

For calculate the probability of getting exactly 4 questions right and 1 wrong, we have five events with their respective probabilities:

1. The wrong answer is the first one:

P=0.75*0.25*0.25*0.25*0.25=0.002929

2. The wrong answer is the second one:

P=0.25*0.75*0.25*0.25*0.25=0.002929

3. The wrong answer is the third one:

P=0.25*0.25*0.75*0.25*0.25=0.002929

4. The wrong answer is the fourth one:

P=0.25*0.25*0.25*0.75*0.25=0.002929

5. The wrong answer is the fifth one:

P=0.25*0.25*0.25*0.75*0.25=0.002929

So the probability of getting exactly 4 questions right and 1 wrong is the sum of probabilities of all the event mentioned:

P=0.002929+0.002929+0.002929+0.002929+0.002929

P=0.014648

Finally the probability of doing well is the probability of having exactly 4 questions right and 1 wrong plus the probability of getting all 5 questions right:

P=0.014648+0.000976

P=0.015624