Friday Night Lights

I was watching Friday Night Lights on TV today. Excellent movie profiling the "economically depressed town of Odessa, Texas and their heroic high school football team, the Permian Panthers" - (IMDB). At the end of the regular football season, there is a 3-way tie between the Panthers and 2 other teams. The coach from each team has to participate in a coin toss to see which 2 out of the 3 teams will compete against each other for a spot in the playoffs. Just in case you haven't seen the movie I won't spoil it any further - but as I was watching this scene I couldn't help but think of some fun Friday Night Lights-inspired probability questions. Here’s one:

Question: If three football coaches each toss one fair coin in succession, what is the probability that at least one of these coins will land on tails?

Solution: Read the question very carefully and make sure you understand what it is asking. Let’s break it down step by step. Let T=tails and H=heads.

Between the three coaches, 1 tail, 2 tails, or 3 tails can be tossed. (Note: we do not care about the circumstance that no tails are tossed because the question specifically asks us to find the probability that AT LEAST one tail is tossed).

If one tail is tossed, it could be the first, second or third coach that made that toss. In other words, there are 3 different options/orders for one tail being tossed:

THH, HTH, HHT

If two tails are tossed, there are three possible combinations for that as well:

TTH, THT, HTT

And if three tails are tossed, there is only one possibility: TTT. Therefore, there are 7 different ways that at least 1 tail can be tossed between the three coaches. Now we need to find the probability of each of these 7 different combinations and then add them all together.

Facts:
-The probability of flipping a tail is ½ and the probability of flipping a head is ½
-Because coin tosses are independent (the result of a coin toss (heads/tails) doesn’t depend on the result of the previous coin toss or on any other factors) we can multiply the probabilities together.

In other words, the probability of THH = (1/2)(1/2)(1/2) = 1/8. The probability of each of the other 6 combinations is also going to be 1/8. So if we add all 7 of these up we get 7/8, which is the answer.

Note: I know this may be a little complicated but the more you practice and become familiar with the different types of probability questions, the easier they will become.

-M