Probability - FAQ

I believe I remember reading that if there is a group of twenty people in a room the odds of two of them sharing the same birthday is less than 50/50. Is this true?

Ginny from Seattle, Washington

The probability of 20 different people all having different birthdays (ignoring leap day) is (364/365)*(363/365)*(362/365)*...*(346/365) = 58.8562%. So the probability of at least one birthday match is 41.1438%. Also, 23 is the fewest number of people needed for the probability of a match to be greater than 50%.

If you have 30 people, all born in the same 365-day calendar year, what is the probability that any two of them will have the same birthday? Please explain the formula in your response.

Scott from Madison, Indiana

Think of the 30 people as lined up. The probability the second person doesn’t match the first person is 364/365. Then, assuming they didn’t match, the probability the next person does not match either of the first two is 363/365. Then keep going one person at a time. The overall probability no two people match is (364/365)*(363/365)*...*(346/365) = 29.3684%. It is often asked what is the fewest people you need for the probability of a match to be at least 50%. The answer is that with 23 people the probability of at least one match is 50.7297%.

There are 75 multiple choice questions in an exam. Each question contains 4 possible answers only 1 is correct. The exam pass mark is 50%. What are the chances of passing the exam by guessing each answer?

Wendy from London

1 in 635,241.

Life expectancy for people of various ages has been calculated and summarized with data at the Social Security web site. However, I want to know the life expectancy of two people. Say I have two people: a thirty-year-old male (me) and a twenty-eight-year-old female (my gf). According to the chart, I will live another 46.89 years and she will live another 53.22 years. But, how long is it expected until we both are dead? How do I calculate this?


First, it would be appropriate to use cohort life tables, as opposed to the period life table you linked to. I tried to find cohort life tables online but was unsuccessful. However, we can still use the table provided. It may underestimate how long you will live slightly, because it won’t take into account future increases in life expectancy.

Answering your question involved creating a large matrix of the probability of each combination of year of death for you and the 28-year-old female. Forgive me if I don’t get into the details. The bottom line is that I show that first one of you will die in 41.8 years, and the latter death will be in 57.3 years. Both figures round down; in other words, you don’t get credit for partial years.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

What is i^i?


I don't want to just tell you the answer without giving you the opportunity to solve it yourself.

First, here is a hint to help. If you don't already know this equation, you're unlikely to solve it.

Otherwise, I admit I'm bad. Just show me the solution.

For discussion about the equation in the hint, please visit my forum at Wizard of Vegas.

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