What Is the Probability That Your Second Daughter Is the Fourth Child?

[bball3212]

Question: What is the probability your 1st daughter is the 4th child?

Attempt:
So probability of either male or female is .5 it seems probability of 3 boys in a row is .5*.5*.5 and then times again by .5 to get the answer (so .5^4). Is that correct? How would it change if you then switch that to the probability of 2nd daughter being the 4th child?

 

[disregardthat]

The probability that the 2nd daughter is the 4th child is the probability of getting two sons and one daugther, and then one daughter.

If S stands for son, and D stands for daughter, the outcomes are

DSSD
SDSD
SSDD

all with the same probabilitiy (1/2)^4. Hence 3/2^4.

 

[HallsofIvy]

disregardthat, the question was "what is the probability that your fourth child is your first daughter?" You have answered "what is the probability that your fourth chile is your second daughter". Should we disregard that?

In order that your fourth child be your first daughter, your first three children must be boys. Assuming that boys and girls are equally likely, the probability of that is $1/2^3= 1/8$. Of course, then, the last child must be a girl and the probability of that is also 1/2. The probability the first four children are boys and the fourth a girl is (1/8)(1/2)= 1/16.

 

[disregardthat]

It was his second question.

 

[dacruick]

The probability the first four children are boys and the fourth a girl is (1/8)(1/2)= 1/16.

HallsofIvy gave you the answer to your first question (minus that typo, he meant first three children).

Using his process, how do you think you can extend that to the second question? How, specifically, does the probability change.

There is a line of thought that is helpful here, which I'll hint you to. There is only one way for the first 3 children you have to be boys, but that is not the case for the second question.