I was reading that the pass rate for a particular test was 60%. That doesn’t sound very high – and if I take that exam I have a 40% chance of failing that test.

But what if I retry? Or retry two times? What are my odds of passing in that situation?

Let’s take the example where the average pass rate of an exam is 60%.

Description |
Math |

Let P_{pass} be the probability of passing |
P_{pass} = 0.6 |

Probability of failing first time is P_{fail} |
P_{fail} = 1 – P_{pass} = 0.4 |

Probability of failing two times is P_{fail}^{2} |
P_{fail_twice} = P_{fail} * P_{fail} = 0.4 * 0.4 = 0.16 |

Probability of not failing two times in a row |
1 – P_{fail_twice} = 1 – 0.16 = 0.84 |

Probability of not failing is the probability of succeeding |
0.84 |

So if you budgeted to take a test two times, not just once, then the probability you pass the first or second time is a much higher 84%!

You could also calculate what your probability of success is if you took an exam three times!

Here’s a table of exams and what your success rate would be if you undertook exams a different number of times:

Description |
10% |
20% |
30% |
40% |
50% |
60% |
70% |
80% |
90% |

one attempt |
10% |
20% |
30% |
40% |
50% |
60% |
70% |
80% |
90% |

two attempts |
19.0% |
36.0% |
51.0% |
64.0% |
75.0% |
84.0% |
91.0% |
96.0% |
99.0% |

three attempts |
27.1% |
48.8% |
65.7% |
78.4% |
87.5% |
93.6% |
97.3% |
99.2% |
99.9% |

four attempts |
34.4% |
59.0% |
76.0% |
87.0% |
93.8% |
97.4% |
99.2% |
99.8% |
100% |

The moral of the story is: don’t fear a tough-looking exam with a mere 60% success rate. Know that if you budget the time and expense for taking the test twice you increase your odds of passing to a much higher 84%!

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