## Why Probabilities Cannot Add Up To One

Consider a good goalscorer playing football. The probability of him scoring a gal in any one minute of play is 0.01.You may think then that in a game of 90 minutes, the probability of him scoring is\[90 \times 0.01=0.9\]

and if extra time needs to be played - an extra 30 minutes - the probability would be \[120 \times 0.01=1.2\]

.At this point your naivety is exposed. YOU CANNOT HAVE A PROBABILITY MORE THAN ONE!

To see why, consider the equation

\[P(He \: scores)+P(He \: does \: not \: score)=1\]

Rearranging this gives

\[P(He \: scores)=1-P(He \: does \: not \: score)\]

If we assume the probability of scoring in each minute is independent from minute to minute (and he cannot score twice in a minute), the probability of him not scoring in a minute is

\[1-0.01=0.99\]

and the probability of him not scoring in 90 minutes is \[0.99^{90}=0.4047\]

.The probability of him scoring in 90 minutes is

\[1-0.4047=0.5953\]

.In fact we can draw up the table, which shows how the probability of him having scored in the first so many minutes increases as the game winds ob, with the rate of increase decreasing.

Minutes | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 |

P(He has Scored) | 0.0956 | 0.1821 | 0.2603 | 0.3310 | 0.3950 | 0.4528 | 0.5052 | 0.5525 | 0.5953 |