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.

Guest Dec 6, 2019

#1**+1 **

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.

Guest Dec 6, 2019

#4**+1 **

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.

Guest Dec 8, 2019