Two things at once

What is the chance of flipping heads AND rolling a six? Of two coins both landing on heads? Of taking two red marbles from a bag? Questions about two things at once are where most people start to guess, and where guessing goes wrong. Even a famous French mathematician, Jean d'Alembert, got a question about two coin flips wrong in 1754. There are two plain methods that always work: list every result, or multiply the chances. This page shows both, and when each one is the right tool.

Method 1: list every result

Flip two coins. To see every result clearly, imagine that one coin is gold and the other is silver. Write H for heads and T for tails, gold coin first.

There are 4 results, and they are equally likely. Now every question is a counting question:

• Both heads: only HH, so 1/414.
• Exactly one head: HT and TH, so 2/4 = 1/224 = 12.
• At least one head: everything except TT, so 3/434.

Three coins give 2 × 2 × 2 = 8 results: HHH, HHT, HTH, THH, HTT, THT, TTH and TTT. Exactly two heads is HHT, HTH and THH, so 3/838. At least two heads adds HHH, so 4/8 = 1/248 = 12.

Method 2: multiply the chances

Two events are independent when one cannot change the other. A coin does not know what a die shows. Two dice do not know about each other. For independent events there is a short rule.

Why multiply? Flip a coin and roll a die. Heads happens half of the time. And in that half, a 6 happens one sixth of the time. One sixth of one half is 1/12112. You can also count: a coin and a die give 2 × 6 = 12 results, and only "heads and 6" works.

To multiply two fractions, multiply the tops and multiply the bottoms: 2/3 × 3/4 = 6/12 = 1/223 × 34 = 612 = 12.

Two dice

Two dice give 6 × 6 = 36 equally likely results. Think of one die as red and one as blue: a red 2 with a blue 5 is a different result from a red 5 with a blue 2.

For "both dice show the same number" (a double), the six results 1-1, 2-2, 3-3, 4-4, 5-5 and 6-6 work. That is 6/36 = 1/6636 = 16. Here is a quicker way to see it: the first die can show anything, and the second die must match it, which happens 1 time in 6. In the same way, "different numbers" is 5/656, because the second die misses the first one in 5 of its 6 faces.

Marbles: putting back or not

Take a marble from a bag, then take a second one. The answer depends on one thing: did you put the first marble back?

With putting back (also called "with replacement"), the bag is the same both times. The two draws are independent, and the chance is the same fraction twice.

Without putting back, the bag changes. After you take one red marble, there is one marble fewer in the bag, and one red marble fewer. So for the second draw, the top and the bottom of the fraction both go down by 1.

Coins and dice have no memory

A coin lands on heads 5 times in a row. Is tails now "due"? No. The coin does not remember what it did before. The next flip is still heads with a chance of 1/212, and a die is still a 6 with a chance of 1/616, whatever came before. Believing that the past changes the next result is called the gambler's fallacy (a fallacy is a wrong idea that sounds right). A famous story comes from a casino in Monte Carlo in 1913. The roulette ball landed on black 26 times in a row, and it is said that people lost a lot of money betting that red had to come next.