Question

A supposed coffee connoisseur claims she can distinguish between a cup of instant coffee and a cup of drip coffee 75% of the time. You give her 6 cups of coffee and tell her that you will grant her claim if she correctly identifies at least 5 of the 6 cups.

(a) What are her chances of having her claim granted if she is in fact only guessing

(b) What are her chances of having her claim rejected when in fact she really does have the ability she claims?

[Round your answer to 4 decimal places.]

Answer 1

I would try as follows:
a) if she is just guessing, the the possibility of being right is 50%, and 50% for being wrong (since these are the only choices) in each trial. her claim will be granted in two cases: either she gets 5 right out of 6 or 6 out of 6, so
P(her claim granted)= P(being right 5 times)+P(being right 6 times)=0.5^5*0.5+0.5^0.5=1/32

Answer 2

1/32=0.0313

Answer 3

b) in this case (she has the ability that she's claims), P(being right)=0.75 and P(being wrong)=0.25..
her claims will be rejected if she gets less than 5 right answers, either 4,2,3,1 or 0 right answers.. but this is the same as 1-P(her claim granted), hence
P(her claim rejected)=1-P(her claim granted)=1-P(5 right answers)-P(6 right answers)=
1-0.75^5*0.25-0.75^6=1-0.237=0.763