The quiz consists of 3 problems on algebra and 3 on geometry. Each problem has 4 choices with only 1 correct. The student answers them randomly. Find the probability

Question

a) that the number of correct answers in algebra is larger than in geometry by 1

b) to have at most a score of 40 if a correct answer in algebra is worth 10 and in geometry is 15, wrong answer is 0

Answer 1

The answers in algebra and geometry can be considered independent.

The density function of algebra and geometry answers is exactly the same; binomial with p=.25.

The joint density function of the answers is simply the product of the two density functions.

Make a 2 way table (like a spreadsheet) with the joint density function; it will be a symmetric table.

Then add up the density of each space in the table which satisfies the conditions in part a).

In part b, use the same table but attach weights to your sums (10 for a correct answer in algebra and 15 for a correct answer in geometry).

This is an exercise in arithmetic as well a probability question...BE CAREFUL!

Added later

The probabilities required all have denominator 64² and the joint density function is not too hard to prepare.

The first answer has only 3 terms and the result is 981/4096, a little less than 1/4.

The second answer has 9 terms...and I didn't bother to do the addition.