Question about calculating odds

[Miss Chicken]

Myself and a friend were discussing this today and, as I haven't had any mathematics classes for years and years, I am not sure how to calculate the odds of this.

I have two sisters. The first has 4 children born in 1977, 1979, 1983 and 1985. The second has three children born in 1979, 1981 and 1983. Each of my sisters' youngest child was born on my birthday. I am trying to work out what are the odds of having 2 out of 7 of my nieces/nephews born on my birthday?

 

[Skellington]

I think a rough calculation works out at 7/365 * 6/365, or about 1 in 3,172. That's ignoring the extra day in leap years and assuming that the birth dates are random (with your own as a constant).

 

[Miss Chicken]

Thank you. That makes perfect sense.

I did it wrong. I divided 365 by 7 and then multiplied the answer by itself - 7/365 * 7/365.

my friend tried to tell me I had to take the fact that they were born over an 8 year period into account. I I kept telling her 'I don't think so' though I did mention that there were two leap days during 1977 and 1985.

 

[Trekker4747]

Sounds like a classic case of the Birthday Paradox.

 

[Robert Maxwell]

Sounds like a classic case of the Birthday Paradox.

Yup, that's exactly what I was going to post. Our intuition about this sort of problem is very wrong and the reality is much weirder.

This is more of a special case of the birthday problem and I will post more about that later, since it's kind of complicated.

 

[Robert Maxwell]

Taking it as a classic birthday problem, this means we have a set of 8 people (you plus your nieces and nephews) of which 3 share birthdays (you and two of the kids.) For this, we will ignore distribution effects (in reality, more people are born on certain dates than others) and assume all 365 days are equally likely.

Let's use the Poisson approximation. We want this event to occur 3 times in a set of 8 and the event has a 1/365 chance of occurring per chance of event. You can use the calculator here.

Assuming there is an equal chance of being born on any particular day (which there isn't), the odds of having 3 people out of 8 born on the same day is 0.00017168064412993736%, which is very tiny.

If you'd like a more accurate number, we'd need to know your birthday.

 

[Green Shirt]

Wouldn't a group of couples, attempting to conceive on the same date somehow skew the results? Of course, not every woman in a group who conceives on the same date will necessarily give birth exactly nine months later.

It might make an interesting experiment, or not.

 

[Miss Chicken]

My birthday is in February which, because it is the shortest month, tends to be the month that the fewer births occur no matter where in the world one lives.

I did find a site which lists how many babies where born in each month in Australia in 2009.

http://www.fisherphotography.com.au/what-month-are-most-babies-born/

Taking it as a classic birthday problem, this means we have a set of 8 people (you plus your nieces and nephews) of which 3 share birthdays (you and two of the kids).

If I include myself shouldn't I include my 3 siblings as well?

Edit to add - and my three sons as well, which will make up all 14 people in these two generations of my intermediate family.

Of those 14

5 were born in February (all on or after the 18th of February)
3 in August
2 in September
1 in January (in fact the last day of the month)
1 in June
1 in July
1 in October

 

[Jedi_Master]

[Han Solo] Never tell me the odds [/Han Solo]

 

[Trekker4747]

Taking it as a classic birthday problem, this means we have a set of 8 people (you plus your nieces and nephews) of which 3 share birthdays (you and two of the kids.) For this, we will ignore distribution effects (in reality, more people are born on certain dates than others) and assume all 365 days are equally likely.

Let's use the Poisson approximation. We want this event to occur 3 times in a set of 8 and the event has a 1/365 chance of occurring per chance of event. You can use the calculator here.

Assuming there is an equal chance of being born on any particular day (which there isn't), the odds of having 3 people out of 8 born on the same day is 0.00017168064412993736%, which is very tiny.

If you'd like a more accurate number, we'd need to know your birthday.

I'm not entirely sure if I "get" why some days would be more likely than others for a birth to occur. I would think every day of the year has an equally likely chance for a birth to happen. I mean, people *are* born everyday and people do have sex everyday and if that intercourse results in a pregnancy and if everything goes well during gestation a birth will occur 9-10 months later.

There are the occasional "baby boom" events centered around national events, local events, natural disasters, Valentine's day, people's own birthdays, stuff like that and, naturally, nine months after an "event day" will see an increased likelihood of births but it seems like there's terribly a lot of variables in there I think it'd be hard to say what day(s) are more likely to have births occur over others. It may not be that every single day of the year is equally likely but I'd think the variation isn't too steep beyond maybe October/November seeing the largest "boom" given that a birth then would be a result of pregnancy occurring in February.

If you want to talk about odds an likelihoods of people sharing birthdays, my neighbors growing up had two daughters and their ages are precisely (to the day, that is) two years apart. They both were born naturally (i.e. the mother didn't have labor induced or a c-section to "force" both children to share a birthday) on the exact same day two years apart.

 

[Robert Maxwell]

Taking it as a classic birthday problem, this means we have a set of 8 people (you plus your nieces and nephews) of which 3 share birthdays (you and two of the kids.) For this, we will ignore distribution effects (in reality, more people are born on certain dates than others) and assume all 365 days are equally likely.

Let's use the Poisson approximation. We want this event to occur 3 times in a set of 8 and the event has a 1/365 chance of occurring per chance of event. You can use the calculator here.

Assuming there is an equal chance of being born on any particular day (which there isn't), the odds of having 3 people out of 8 born on the same day is 0.00017168064412993736%, which is very tiny.

If you'd like a more accurate number, we'd need to know your birthday.

I'm not entirely sure if I "get" why some days would be more likely than others for a birth to occur. I would think every day of the year has an equally likely chance for a birth to happen. I mean, people *are* born everyday and people do have sex everyday and if that intercourse results in a pregnancy and if everything goes well during gestation a birth will occur 9-10 months later.

There are the occasional "baby boom" events centered around national events, local events, natural disasters, Valentine's day, people's own birthdays, stuff like that and, naturally, nine months after an "event day" will see an increased likelihood of births but it seems like there's terribly a lot of variables in there I think it'd be hard to say what day(s) are more likely to have births occur over others. It may not be that every single day of the year is equally likely but I'd think the variation isn't too steep beyond maybe October/November seeing the largest "boom" given that a birth then would be a result of pregnancy occurring in February.

If you want to talk about odds an likelihoods of people sharing birthdays, my neighbors growing up had two daughters and their ages are precisely (to the day, that is) two years apart. They both were born naturally (i.e. the mother didn't have labor induced or a c-section to "force" both children to share a birthday) on the exact same day two years apart.

Births are significantly seasonal. It's not totally nuts--it's not like one month is hugely lopsided vs. another--but there is a very noticeable variance in distribution of birthdays by time of year. The most births happen in late summer/early autumn in the northern hemisphere, which lines up perfectly with people having more sex in the winter. The fewest births happen in the deep winter, as people are having less sex during the hottest months. (The reasons for both should be pretty obvious.)

Now, this seems to apply most to developed countries where people are generally shielded from the elements and have regular access to birth control. I found this study of a rural area in India, in which the distribution is very pronounced: August has almost 3 times as many births as April. These were their key observations:

• The peak period for successful conception appears to be during the winter season (November-December)

• The seasonality of births did not vary between different socioeconomic strata.

• There appears to be a baseline biological seasonality which can be potentiated or inhibited by external factors like use of contraception, socio-cultural and climatic factors.

• If interventions to decrease the birth rate, I like increasing the use of contraception, are scheduled in the months of November-January, they may be more effective.

So, suffice it to say, while there's no reason any given individual couldn't have been born on any day of the year, statistical rates of birth clearly favor summer/autumn months over winter/spring months, and how pronounced the deviation is seems to depend primarily on the availability of contraception.

 

[Bisz]

Assuming there is an equal chance of being born on any particular day (which there isn't), the odds of having 3 people out of 8 born on the same day is 0.00017168064412993736%, which is very tiny.

I don't think that is correct. Based on how Chicken asked her question you are not looking for the chances of 3 out of 8 people having the same birthday. You are looking for the probability of 2 of her 7 nieces and nephews having the same birthday as her.

Using the same assumptions you mentioned the chances of one person having the same birthday as her is 1/365. For two people it is 1/365 * 1/365. Then you have to make sure the other 5 are not born on the same day, so 5 * 354/365. You and (*) the two,

Which gives a result of 0.0037%.

I think. I always hated finite.

 

[Bisz]

Which gives a result of 0.0037%.

I think. I always hated finite.

In a display of why I dislike finite, it occurred to me, just before falling asleep, the above is not correct. It forces the two children with the same birthday as Chicken to be the first two children, where we are looking for the probability of any two children. Which means we need to multiply by 7 choose 2, i.e. 21.

Which gives the probability as 0.08%