What Is the Probability of Three Girls Given the Youngest Is Female?

[annie122]

i got the answer to the following problem wrong:
"there are four children in in the family. what is the probability that there are three girls, given that the youngest child is female?"

my (updated) answer:
the youngest is female, so three out of two children must be female. there are three ways of this happening, (=3C2) so, the answer is (1/2) ^3 * 3 = .375

also, how do i determine if two variables are positively correlated?

 

[chisigma]

Re: two probabilities question

i got the answer to the following problem wrong:
"there are four children in in the family. what is the probability that there are three girls, given that the youngest child is female?"

my (updated) answer:
the youngest is female, so three out of two children must be female. there are three ways of this happening, (=3C2) so, the answer is (1/2) ^3 * 3 = .375

also, how do i determine if two variables are positively correlated?

Lets suppose that the probability of child male and child female is the same, i.e. $p = \frac{1}{2}$. If no information is allowable, then the probability to have three girls and one boy is... $\displaystyle P = \binom{4}{3}\ \frac{1}{16} = \frac{1}{4}\ (1)$

However if You know a priori that one is famale, the probability to have three girls and one boy is the probability to have two girls and one boy among the remaining childs and it is... $\displaystyle P = \binom {3}{2}\ \frac{1}{8} = \frac{3}{8}\ (2)$ Kind regards $\chi$ $\sigma$

 

[HallsofIvy]

Re: two probabilities question

Here is another way to do this: writing "G" for "girl", "B" for "boy", in order from youngest to oldest we could have
GGGG
GGGB
GGBG
GGBB
GBGG
GBGB
GBBG
GBBB
The first letter is always "G" because we are told that the youngest child is a girl. The others have $$2^3= 8$$ possible orders giving 8 possible situations. Of those 8, exactly three have 3 "G" (GGGB,, GGBG, GBGG). Assuming that boys and girls are equally likely the probability of "three girls" is 3/8= 0.375.

(If the problem were "at least three girls" we would include "GGGG" so the probability would be 4/8= 0.5.)

 

[annie122]

Re: two probabilities question

thanks, I'm cleared now :)

 

[CellCoree]

As a scientist, it is important to understand the concept of probability and how to apply it in various situations. In this particular problem, the key information given is that the youngest child is female. This means that out of the four children, one must be a female. Therefore, the probability of having three girls in the family is dependent on the remaining three children.

To calculate the probability, we can use the formula for conditional probability, which states that the probability of event A occurring given event B has already occurred is equal to the probability of both events occurring divided by the probability of event B occurring.

In this case, event A is having three girls in the family and event B is the youngest child being female. The probability of event A is equal to the number of ways three girls can be chosen out of four children, which is 4C3. The probability of event B is simply 1, since we already know that one child is a girl.

Therefore, the probability of having three girls in the family given that the youngest child is female is (4C3)/1 = 4/1 = 4.

To determine if two variables are positively correlated, we need to look at their relationship. A positive correlation means that as one variable increases, the other variable also increases. This can be represented by a positive slope on a scatter plot.

To determine if two variables are positively correlated, we can calculate the correlation coefficient, which is a measure of the strength and direction of the relationship between two variables. A positive correlation coefficient indicates a positive correlation between the two variables. Additionally, visualizing the data on a scatter plot can also help determine if there is a positive correlation between the two variables.