Problem 3. Tom has one sibling. What is the probability that he has a sister.
Suppose I tell you that the Smiths have 2 children, and one is a boy. Then the probability that the other is a girl is indeed 2/3 by the previous reasoning.
Now suppose that I tell you that the Smiths have two children, and one of them, Tom, is a boy. Now the probability of the other sibling being a girl is 1/2 by your reasoning that the other child can be male or female.
The difference is in your list of possibilities: MM,MF,FM. The problem is the MM pair -- in the case that the two are indistinguishable, the probability is 2/3, but in the case that they are distinguishable there are two cases -- Tom being older or younger which makes the probability 1/2.
Here is another case where prior knowledge makes a difference:
In the classic Monte Hall problem, the contestant picks one of three doors. Monte opens a losing door, and asks the contestant if she wants to switch her choice. She wins 2/3 of the time if she switches. Now suppose her husband, who has been backstage and could not hear what has happened, is brought on stage. He sees that there are now two choices: he would have a 1/2 chance of finding the winning door.
If this question had asked the probability of a pair of siblings having a girl if one of the siblings is known to be a boy then the answer would be 2/3. This is because the possibilities are BB,BG, or GB.
However, this question specifically points out one of the siblings.
Now Tom can be the older sibling or the younger sibling. If the older, he can have a younger brother or a younger sister. If he is the younger sibling, he can have an older brother or an older sister. Let T be Tom, B be a brother and S be a sister. The order of the pairs is the birth order:
The possibilities are (T,B),(T,S),(B,T),(S,T). There are 4 cases, two of which have a female sibling, so the correct answer is indeed 1/2.
The reason for the difference is that there are now 2 different cases of two boys. If M is male and F female, we do not have MM,MF,FM,FF. Instead we have Mm,mM,MF,FM where M,m and m,M are the different cases of whether Tom is older or younger.
There is mathematical answer, and there is also a real-life answer.
It could be dependent on how old Tom's sibling is. According to the U.S. Department of Labor, newborn boys outnumber girls 51% to 49%.
However, over the longitude of time, that number changes. According to the U.S. Census Bureau, boys outnumber girls in youth. As the population ages, females begin to outnumber men around the age of 35-40. This holds true for both the U. S. and the world in general.
However, with the population growth, and the number of younger people in this world, currently, males make up 50.35% of the world population.
So, based on this information, I would contend that Tom's sibling is more likely to be male.
Here is another way to think of this:
(1) Suppose you flip two indistinguishable coins and you are told that one of the coins is heads. Then the probability that the other coin is tails is 2/3 as the possibilities were HH,HT,TH.
(2) Suppose we flip a quarter and a dime. You are told that the quarter is heads. Then the probability that the dime is tails is 1/2. This corresponds to the actual question as Tom is distinguishable from the other male.
qHdH,qHdT,qTdH,qTdT are the only possibilities and only the first two have quarter heads. This is different from HH,HT,TH,TT because now the coins would be indistinguishable.
This question can be related to that of a coin, a coin has 2 sides, therefore it's probability is 0.2, in the same way there are 2 genders (male or female) so it can either be one or the other therefore it's probability is also 0.2.
The possible outcomes are (M,F),(M,M). Where M stands for Male and F stands for Female.
Probability questions can be broken up into 2 steps:
First, how many possible outcomes could there be. In this situation, there are 2 possible outcomes, either Tom has a brother or a sister since he only has one sibling. This makes up the denominator of your probability ratio.
Second, determine how many chances are possible for the asked situation to happen (example: 3 red marbles, 2 blue = 3 chances to pull red). In this case there is only one chance that Tom has a sister, this makes up the numerator of your probability ratio.
Therefore, the probability of Tom having a sister is 1/2, or 50%.
Its pretty simple since theres only 2 type of genders (Male and Female) so the probability is 1 of 2.
it levels out to around 1/2 because the other child can be either a girl or a boy.