MHB Variation of the shared birthday problem

  • Thread starter Thread starter Asifbymagic
  • Start date Start date
  • Tags Tags
    Variation
AI Thread Summary
In a team of fourteen, the challenge is to calculate the probability of two pairs of members sharing the same birthday. The initial approach involves counting unique birthdays for ten members and then selecting pairs from the remaining four. The calculation considers the total ways to assign birthdays and the combinations of non-paired individuals. The resulting probability is approximately 6.26 x 10^-3. This problem illustrates the complexities of probability in scenarios involving multiple shared birthdays.
Asifbymagic
Messages
1
Reaction score
0
I work as part of a team of fourteen. No big challenge to work out the probability of two of us sharing a birthday. It's a well-documented puzzle.

In my team, we've gone one better: we have two dates where two people have birthdays on that day!

Trying to work out the probability has us stumped. Not of 4/14 not having unique birthdays, but of two pairs of shared birthdays.

Can someone work me through a solution?

Thanks! Alex
 
Mathematics news on Phys.org
I am not an expert in probability, but I've studied it some. Here's my approach.

Let's try counting, and let's start with the $10$ folks with unique birthdays. There'd be $365$ ways to choose the first birthday, $364$ for the second, and so on (ignoring leap year). That is, you'd have $\dfrac{365!}{355!}$ ways to choose these birthdays. Now let's choose the first pair. There are $355$ ways to choose the first pair's birthday, and thus $354$ ways to choose the second pair's birthday. But this is simply reasoning the same way as before, so you have $\dfrac{365!}{353!}$ ways to choose everyone's birthday. However, this number, as is, is too restrictive, because the way I've counted so far assumes which people would be in the pairs. We need to count how many ways we can get the pairs. An easier way of doing this is to count how many ways we can get the ten non-pairs. That's simply $\displaystyle\binom{14}{10}$. There are $365^{14}$ total ways to assign the birthdays randomly. So, our probability would be
$$\frac{\displaystyle\left(\frac{365!}{353!}\right)\binom{14}{10}}{365^{14}}\approx 6.26\times 10^{-3}.$$
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top