Probability

If you flip a coin, what is the chance of getting heads? That’s easy: 50/50. In the language of probability, we say that the probability is $\frac{1}{2}$ . That is to say, half the time you flip coins, you will get heads.

So here is a harder question: if you flip two coins, what is the chance that you will get heads both times ? I asked this question of my son, who has good mathematical intuition but no training in probability. His immediate answer: $\frac{1}{3}$ . There are three possibilities: two heads, one heads and one tails, and two tails. So there is a $\frac{1}{3}$ chance of getting each possibility, including two heads. Makes sense, right?

But it is not right. If you try this experiment 100 times, you will not find about 33 “both heads” results, 33 “both tails,” and 33 “one heads and one tails.” Instead, you will find something much closer to: 25 “both heads,” 25 “both tails,” and 50 “one of each” results. Why?

Because hidden inside this experiment are actually four different results , each as likely as the others. These results are: heads-heads, heads-tails, tails-heads, and tails-tails. Even if you don’t keep track of what “order” the coins flipped in, heads-tails is still a different result from tails-heads, and each must be counted.

And what if you flip a coin three times? In this case, there are actually eight results. In case this is getting hard to keep track of, here is a systematic way of listing all eight results.

When you make a table like this, the pattern becomes apparent: each new coin doubles the number of possibilities. The chance of three heads in a row is $\frac{1}{8}$ . What would be the chance of four heads in row?

Let’s take a slightly more complicated—and more interesting—example. You are the proud inventor of the SongWriter 2000tm.

The user sets the song speed (“fast,” “medium,” or “slow”); the volume (“loud” or “quiet”); and the style (“rock” or “country”). Then, the SongWriter automatically writes a song to match.

How many possible settings are there? You might suspect that the answer is $3+2+2=7$ , but in fact there are many more than that. We can see them all on the following “tree diagram.”

If you start at the top of a tree like this and follow all the way down, you end up with one particular kind of song: for instance, “fast loud country song.” There are 12 different song types in all. This comes from multiplying the number of settings for each knob: $3\times 2\times 2=12$ .

Now, suppose the machine has a “Randomize” setting that randomly chooses the speed, volume, and style. What is the probability that you will end up with a loud rock song that is not slow? To answer a question like this, you can use the following process.

1. Count the total number of results (the “leaves” in the tree) that match your criterion. In this case there are 2: the “fast-loud-rock” and “medium-loud-rock” paths.
2. Count the total number of results: as we said previously, there are 12.
3. Divide. The probability of a non-slow loud rock song is 2/12, or 1/6.