Probability pls help : D

Question

Hannah is taking a 6-item examination. Let her score be denoted by the variable X. Each item is worth 5 points. The exam is graded under the right-minus-wrong rule, which means incorrect and blank items are given a deduction of 5 points per item. Construct the probability mass function for X.

Answer 1

Correct Incorrect/blank X

0 6 - 30

1 5 5 - 25 = - 20

2 4 10 - 20 = - 10

3 3 15 - 15 = 0

4 2 20 - 10 = 10

5 1 25 - 5 = 20

6 0 30

So possible values of X are -30, -20, 10, 0, 10, 20, 30

To find the pmf we have to assign a probability to each of these 7 values

For that we need some probability information which isn't given, specifically what is the probability of a correct answer. Let's call this p

Then the PMF would be

-30 q⁶

-20 C (6,1) p q⁵

-10 C (6,2) p²q⁴

0 C(6,3) p³ q³

10 C(6,4) p⁴ q²

20 C(6,5) p⁵ q

30 p⁶

^{Here q = 1-p}