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1. We say that a quadrilateral is a bow-tie if two of the sides cross each other. An example is shown below. Seven different points are chosen on a circle. We draw all $$\binom{7}{2} = 21$$ chords that connect two of these points. Four of these 21 chords are selected at random. What is the probability that the four chords form a bow-tie quadrilateral?

2. I have a bag with 5 pennies and 6 nickels. I draw coins out one at a time at random. What is the probability that after 4 draws I have removed no more than 2 pennies from the bag?

3. The Bad News Bears are playing against the Houston Toros in a baseball tournament. The first team to win three games wins the tournament. The Bears have a probability of $$\frac{2}{3}$$ of winning each game. Find the probability that the Bears win the tournament.

Dec 6, 2019

#1
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1. The probability is 3*C(7,3)/C(21,4) = 1/57.

2. The probability is (24*1 + 18*2)/C(11,4) = 2/11.

3. The probability is (2/3)^3 + 1/2*1/2*(2/3)^3 = 10/27.

Dec 6, 2019
#3
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Unfortunately, all of these answers seem to be wrong.

Guest Dec 6, 2019
#4
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1. We can count that there are 3*C(7,5) = 63 quadrilaterals that works, so the probability is 63/(4*C(11,4)) = 21/440.

2. The cases are where we draw 4 pennies and 0 nickels, and 1 nickel and 3 pennies.  This leads to a probability of (6*5 + 18*10)/C(11,4) = 7/11.

3. We must consider strings like BBB and BBTB.  Going through the cases, we get a probability of (2/3)^3 + 3*(2/3)^3*(1/3)^2 + 6*(2/3)^3*(1/3)^4 = 304/729.

Dec 8, 2019
#5
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I have not looked and there for do not know if these answers are correct.

However this question has been answered before quite recently. If you search you can probably find the other answers.

Dec 8, 2019