Here’s another example: many fish also lay thousands of eggs. Here are two examples:

This is the largemouth bass. The males scoop out nests with their tails, guard the embryonic fish, and after hatching, herd the young fish around in schools until they are about three weeks old. | |

This is the yellow perch. This species does not provide a nest or any parental protection. Instead, females are escorted by two or more males into shallow weedy areas, where they drape their eggs in an accordion-like strand over the vegetation. The males fertilize the eggs as they are released, and then abandon them. |

Both species can lay 10,000 or more eggs, but the young of the largemouth bass survive somewhat better.

Let’s say that daily survival rates are 99% for largemouth bass and 97% for yellow perch. (Note to all ichthyologists out there: I made up these last numbers, but the stuff above is all true). What is the survival of each species after 21 days (when the young bass are kicked out of the family school)?

Before you read the answer below, please take some time to try to figure it out on your own (you will need a calculator or a spreadsheet, or you could just write down the equation but not do the actual calculations). One of the most important skills you can get from these modules is the ability to extract mathematically useful equations from a big mess of biological information.

OK, lecture over.

Let’s build up slowly. For largemouth bass, the chance of surviving to day 2 is 0.99. The chance of surviving to day 3 is:

P(survive to day 3) = P(survive from 1 to 2 AND survive from 2 to 3) = 0.99 × 0.99 = 0.98

**Likewise, the chance of surviving to days 4 and 5 are:**

P( 1 to 2 AND 2 to 3 AND 3 to 4) = 0.99 × 0.99 × 0.99 = 0.97

P( 1 to 2 AND 2 to 3 AND 3 to 4 AND 4 to 5) = 0.99 × 0.99 × 0.99 × 0.99 = 0.96

**Another way to say this is:**

P(survival to day 3) = 0.99^{2}

P(survival to day 4) = 0.99^{3}

P(survival to day 5) = 0.99^{4}

**See a pattern? So, using a calculator, we get:**

P(largemouth bass surviving to day 21) = 0.99^{20} = 0.82

P(yellow perch surviving to day 21) = 0.97^{20} = 0.54

Even a small difference in daily survival pays off over time.