a radio program has a quiz consisting of 3 multiple choice questions, each with 3 choices. A contestant wins if he or she gets 2 or more of the questions right. The contestant answers randomly to each question. What is the probaility of winning?
p right = .5
1-p right = .5
n = 3
r = 2
C(3,2) = 3!/[ 2!(3-2)!] = 3
P(3,2) = C(3,2) .5^2 .5^1
= 3 * .5^3
prob(right) = 1/3
prob(wrong) = 2/3
prob(2 of 3 correct)
= C(3,2) (1/3)^2 (2/3)
= 3(1/9)(2/3)
= 2/9
prob(getting all 3 correct)
= (1/3)^3 = 1/27
prob(win) = 2/9 + 1/27 = 7/27
checking:
prob(none right) = (2/3)^3 = 8/27
prob(one right) = 3(1/3)(2/3)^2 = 4/9
from above:
prob(two right) = 2/9
prob(3 right) = 1/27
they should add up to 1
8/27 + 4/9 + 2/9 + 1/27 = 27/27 = 1
All is good.
oh, forgot to add prob of getting all three
which is .5^3
so total = 4 * .3^3
Also the prob of a choice being correct is 1/3 not 1/2
Sorry, missed that sentence
To calculate the probability of winning, we need to consider all the possible combinations of correct and incorrect answers the contestant can give.
Let's analyze each possibility:
1. If the contestant answers all three questions correctly:
- The probability of answering one question correctly is 1/3.
- The probability of answering all three questions correctly is (1/3) * (1/3) * (1/3) = 1/27.
2. If the contestant answers two questions correctly and one question incorrectly:
- The probability of answering one question incorrectly is 2/3.
- The probability of answering two questions correctly and one question incorrectly (in any order) is the sum of the following possibilities:
- (1/3) * (1/3) * (2/3) = 2/27
- (1/3) * (2/3) * (1/3) = 2/27
- (2/3) * (1/3) * (1/3) = 2/27
3. If the contestant answers all three questions incorrectly, the probability is:
- The probability of answering one question incorrectly is 2/3.
- The probability of answering all three questions incorrectly is (2/3) * (2/3) * (2/3) = 8/27.
Therefore, the probability of winning is the sum of the probabilities from cases 1 and 2:
(1/27) + (2/27 + 2/27 + 2/27) = 7/27.
So, the probability of winning the radio program quiz is 7/27, which is approximately 0.259 or 25.9%.