A test contains 10 true/false and 5 four-choice questions. If a student has guessed on all the questions, what is the probability of getting a 100%?

Question

how do i figure out my sample space and my answer from there?

Answer 1

On each of the true/false questions, the student has a 1/2 or 0.5 chance of guessing correctly. Since the questions are all independent, the probability of guessing all of them right would be (0.5)^10 = 1/1024

 

Similarly the odds of guessing correctly on any four-choice problems is 1/4. Since there are five of those the probability would be (1/4)^(5) = 1/(4^5) = 1/1024

 

Since the true/false questions are independent of the multiple choice problems, you would multiply the two answers. So final answer would be (1/1024)*(1/1024) = (1/1024)^2 = 1/ (1024^2) = 1/ 1048576

 

Now to see this better, remember that probability problems can be much easier to understand when you look at the simplified version and then once it makes sense you can apply it to more complicated problem. For example let's look at the case where there are only two true/false questions (instead of 10) and one 4-choice problem. The student guesses on all what is the probability of getting 100%?

 

well let's write all the possible combinations first:

 

TTA,TTB,TTC,TTD,TTE,TFA,TFB,TFC,TFD,TFE,FTA,FTB,FTC,FTD,FTE,FFA,FFB,FFC,FFD,FFE

 

As you can see, there are 20 possibilities. But only one of these would get you a score of 100%. So the probability is 1/20.

 

Looking at it another way, there are two possibilities for EACH of the true/false questions, and 5 for EACH multiple choice question. So total number of possibilities would be: 2*2*5=20

 

Now out of the 20, ONLY ONE would give you a score of 100%. So the answer is 1/20.

 

Hope that helps

Answer 2

Total sample space is 2^10*4^5=32*32*256

 

There is only one right answer for each problem so it's one over all that.

 

Hope that helps,

Deanna