Short answer: 1/128, or 0.78125% Show Longer answer: Somewhere between 1/256 and 100%, depending on what you mean by your question. First thing's first - math is excellent at answering specific questions, but it still requires a lot of creativity and skill to know what question to ask. I'm not sure what you mean by your question, and there's a lot of interesting questions (with interesting answers) that this picture could inspire:
All of those are very different questions with very different answers! One last pedantic thing: I'm making the following assumptions:
Each of those assumptions is either possibly wrong, definitely a little bit wrong, or definitely a lot a bit wrong. Anyways, enough ranting about pedantic garbage, let's talk about them odds! What are the odds of having that exact family structure (1 daughter and then 7 sons)? Each child has an independent 50% chance of being born male or female, and the ordering is important - the first child must be female, and the next seven must be male. We find this by taking P(first_child) * P(second_child) * P(third_child)... which ends up being 0.5^8, or 1/256. What are the odds of having exactly 7 sons and 1 daughter? In this case, we don't care if the daughter was first, second, third... we just care that there are eight kids, and seven of them are boys. For this, we take P(first_child) * P(second_child) * P(third_child)... again, which is 0.5^8 or 1/256, but then we multiply it by the number of ways to re-organize the set. In this case, there are 8 ways (putting the daughter in spots 1-8), which gives us 8/256 or 1/32. What are the odds of having only one girl (or boy) with eight kids? Because the odds of having a boy/girl are the same, the 1/32 from above applies both to 7 boys and 1 girl or 1 boy and 7 girls. We can add together the odds of those two mutually exclusive cases to get 2/32, or 1/16. What are the odds of having 7 sons in a row? Easy, we go back to the first answer for this one - 0.5^7, which is 1/128 What are the odds that at least one random Ellis Island immigrant family had 1 daughter (oldest) and 7 sons? The odds of any one family having that exact structure are 1/256. If we assume that there were 5,000 families with 8 kids (a completely random and silly assumption based on a complete guess but sounds reasonable), we can roughly guesstimate this. We have to pull out a weird probability trick here - to find this answer using the same tricks from before, we'd have to...
Thankfully, there's a much easier way - we can calculate the odds that every family had any family structure other than 1 daughter followed by 7 sons, and then say that the answer to our original question is the odds of that case not happening. P(not having 1 daughter and then 7 sons) = 255/256 Applying the trick from the second question, we can raise that to the power of 5000 to apply it to all families... P(no family had 1 daughter and then 7 sons) = 0.000000317%. So the odds of a family like this coming across through Ellis Island is probably somewhere in the ballpark of 99.999999283%, or damn near 100% likely. EDIT: I think it's interesting to also ask what is the odds that this is the ONLY family that came through Ellis Island with this structure. The odds of that are ((1/256) * ((255/256)^4999) * 5000) - the inner part is the odds that the first family of that size that came through had the 1daughter 7sons structure and then the 4999 others did not, and multiply that answer by 5000 because we don't care if they came in first, second, third... and there are 5,000 different possible permutations of the ordering that are valid: 0.0000062%. If my assumptions are correct (which they aren't), there's a 57.44% chance that somewhere between 16-22 such families came through Ellis Island. This should make sense - odds of such a family are 1/256, times 5000 families you'd "expect" 19-20 families (19.53125) to have that size. What is the probability of 3 children being girls?First, since the chance of a child being a girl is 1/2, the chance of all three being girls is (1/2) (1/2) (1/2) = 1/8.
What are the odds of having 4 daughters in a row?Answer and Explanation: The probability of a woman giving birth to four girls in a row is 1/16.
Is having 3 daughters lucky?“Vietnamese tradition says that if you have three girls, you will be lucky for a very long time,” Tran joked Monday as he sat with his wife and two other daughters in a room at Fountain Valley Regional Hospital and Medical Center.
What are the odds of having 5 daughters in a row?P(B): The probability of having at least one boy = 1 - probability of having five girls = 1 - (1/2)⁵ = 31/32.
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Probability of having 3 daughters in a row
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