Question 1182481: A panel of 7 judges is to decide which of 2 final contestants (A and B) in a beauty contest
will be declared the winner (based on a simple majority of the judges) Assume 4 of the
judges will vote for A and the other 3 will vote for B. If 3 of the judges are randomly
selected without replacement, what is the probability that a majority of them in the
sample will favor A?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Case 1) All three judges favor contestant A
The fraction 4/7 is the probability a judge favors A, which is also the probability of selecting such a judge. After that selection is made, 3/6 is the probability of selecting another judge of a similar mindset. Then finally 2/5 is the probability of selecting the third judge who votes for contestant A.
4/7 = probability selecting that first judge
3/6 = probability selecting the second judge
2/5 = probability selecting the third judge
All three judges favor contestant A.
Note that the numerators count down (4,3,2) and the denominators do as well (7,6,5). This countdown is because we are not replacing the judges after they are selected.
We then multiply out those fractions
(4/7)*(3/6)*(2/5) = 4/35 is the probability we select 3 judges who favor contestant A.
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Case 2) Exactly 2 judges selected voted for contestant A.
Let's label the judges as {C,D,E,F, G,H,I}
Judges C through F favor contestant A
Judges G through I favor contestant B
We have 3 ways to pick the judge who favors contestant B
We have (4*3)/2 = 6 ways to pick the two judges who favor contestant A
So overall, there are 3*6 = 18 different outcomes for case 2.
There are 7 C 3 = 35 ways to pick the three judges where order doesn't matter and we don't care who they voted for. I'm using the nCr combination formula (with n = 7 and r = 3).
The probability of getting case 2 to happen is 18/35
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Cases 1 and 2 constitute all the possible outcomes where contestant A is favored over B. In other words, this covers all the bases so that contestant A gets the majority.
Adding those results gets us
4/35 + 18/35
(4+18)/35
22/35
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Answer: 22/35
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