An exam is scored out of 100 points and has a total of 10 questions. The minimum

Question

An exam is scored out of 100 points and has a total of 10 questions. The minimum passing grade is 60 points. Each question is a TRUE/FALSE question. The first 8 questions are worth an equal number of points for a total of 60 points. The last two questions are each worth 20 points. Linus has decided to guess the answer to each question. He will not answer FALSE for any two consecutive questions. In how many ways can Linus answer all 10 questions? Patty has also decided to guess the answer to each question. Given that Patty answers both of the last two questions correctly, what is the probability that she passes the exam? Find the probability that Patty passes the exam.

Explanation / Answer

a) There are two arrangements possible

1) Start with true

T _ T _ T _ T _ T _

If linus guesses 1 false it can be done in 5C1 ways

similarly for 2 false = 5C2

  for 3 false = 5C3

for 4 false = 5C4

for 5 false = 5C5

2) Do not start with true

_ T _ T _ T _ T _ T _

In This case also same arrangement possible as above

Hence total ways are, 2 * ( 5C1 + 5C2 + 5C3 + 5C4 + 5C5)

= 2 * ( 5 + 10 + 10 + 5 + 1)

= 62

B) 1) iF PATTY ANSWERS LAST 2 QUESTIONS CORRECTLY, THIS MEANS SHE SCORED 40

She needs 20 or more to pass the exam

Consider a situation where she fails, this means she answers less than 3 out of 8 correctly

Probability of fail is 8C2 * ( 0.5)^2 * ( 0.5)^6 +  8C1 * ( 0.5)^1 * ( 0.5)^7

= 0.14

Probability of pass = 1 - probability of fail

= 1 - 0.14

= 0.86

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