## U.S. Senator John McCain and Former USN pilot, U.S. Naval Academy Commencement Address, 1993

“As ensigns and second lieutenants, the character of the young sailors and Marines entrusted to your care will be formed in large part by their appreciation of your character. You are where leadership begins. You are the models who stand just past the sergeants and chiefs, and those under your command will derive from your behavior the direction of their own lives. Their firm respect for you, on which their lives and our security will depend, will be determined by how faithfully you keep, on duty and off, the code you learned here.

I will go to my grave in gratitude to my Creator for allowing me to stand witness to such courage and honor. And so will you. My time is slipping by. Yours is fast approaching. You will know where your duty lies. You will know. God bless you. Semper Fi. Fair winds and following seas.”

– – U.S. Senator John McCain and Former USN pilot, U.S. Naval Academy Commencement Address, 1993

## Combinatorics problem

1. You are given a ten piece box of chocolate truffles. You know based on the label that six of the pieces have an orange cream filling and four of the pieces have a coconut filling. If you were to eat four pieces in a row, what is the probability that the first two pieces you eat have an orange cream filling and the last two have a coconut filling?

(given, O = orange, C = coconut) :

P(OOCC)

= P(1st is orange) * P(2nd is orange) * P(3rd is coconut) * P (4th is coconut)

= 6/10 * 5/9 * 4/8 * 3/7

= .6 * .556 * .5 * .429

= .0716

Follow-up question: If you were given an identical box of chocolates and again eat four pieces in a row, what is the probability that exactly two contain coconut filling?

• Step 1 involves a combinatorics problem of 4 choose 2 to determine how many combinations of oranges and coconuts we can obtain given 4 pulls from the box :

= 4C2

= 4!/ (2! * (4-2)!)

= 24 / (2 * 2)

= 6

This is equivalent to the following six combinations:

CCOO, COCO, COOC, OCCO, OCOC, OOCC

• Step 2. In the first question above, we learned that the probability of pulling exactly 2 coconuts in 4 pulls from the box is the same at pulling exactly 2 oranges as well and we can see that the probability is the same for all 6 combinations of Os and Cs.

Therefore, the probability of pulling exactly 2 coconuts

=  4C2 [see step 1 above] * .0716

=  6 * .0716

=  .4296