Question

The first part of a test consists of ten true-false questions. The second part of the test consists of ten multiple-choice questions, each with six alternatives, only one of which is correct. What is the probability of correctly
guessing the answers of: At least six questions in the first part of the test,
and At least six questions in the second part of the test

Answer 1

first part:
prob(right) = 1/2
prob(not right) = 1/2
prob(at least 6 of 10) = C(10,6)(1/2)^6 (1/2)^4 + C(10,7)(1/2)^10 + ... + C(10,10)(1/2)^10 , {the last part is (1/2)^10 for each}
= (1/2)^10[C(10,6) + C(10,7) + C(10,8) + C(10,9) + C(10,10)]
= (1/1024) (210+120+45+9+1)
= 385/1024

2nd part
prob(right) = 1/6
prob(not right) = 5/6
prob(at least 6 right) = C(10,6)(1/6)^6 (5/6)^4 + C(10,7)(1/6)^7 (5/6)^3 + C(10,8)(1/6)^8 (5/6)^2 + C(10,9)(1/6)^9 (5/6) + C(10,10)(1/6)^10
= 210(625/6^10) + 120(125/6^10) + 45(25/6^10) + 9(5/6^10) + 1/6^10
= .002438

prob(your event) = 385/1024 (.002438)
= .00917

you might want to check my arithmetic, should have written it down
as I went along