How is P(GB) equal to the P(G)P(B)?

Question

I have read the Probability and Statistics for Engineering and the Sciences by Jay Devore, and then I got stuck at the example 3.12 (ch3 Discrete Random Variables and Probability Distributions) as following

Starting at a fixed time, we observe the gender of each newborn child at a certain hospital until a boy (B) is born. Let p=P(B), assume that successive births are independent, and define the rv X by x=number of births observed. Then
p(1) = P(X=1) = P(B) = p
p(2) = P(X=2) = P(GB) = P(G)P(B) = (1-p)p
…

I am little curious and can not figure out that how(why) is P(GB) equal to P(G)P(B) ?

Answer 1

The notation could be improved:

• $P(B)=p$ is supposed to be the probability any particular child is a boy, and $P(G)=1- p$ is supposed to be the probability any particular child is a girl.

• $P(GB)$ is supposed to be the probability that the you see a girl first and then a boy second. It might be better written as $P(G_1,B_2)$, i.e. that the first child is a girl and the second child is a boy.

• Using conditional probability, this is $P(G_1,B_2)=P(G_1)P(B_2 \mid G_1)$.

• Since "successive births are independent" you have $P(B_2 \mid G_1)=P(B_2)=p$, as well as $P(G_1)=1-p$. Equivalently, use independence to say $P(G_1,B_2)=P(G_1)P(B_2)$.

• So $P(G_1,B_2) = (1-p)p$.

Answer 2

I have read the Probability and Statistics for Engineering and the Sciences by Jay Devore, and then I got stuck at the example 3.12 (ch3 Discrete Random Variables and Probability Distributions) as following

Starting at a fixed time, we observe the gender of each newborn child at a certain hospital until a boy (B) is born. Let p=P(B), assume that successive births are independent, and define the rv X by x=number of births observed. Then p(1) = P(X=1) = P(B) = p p(2) = P(X=2) = P(GB) = P(G)P(B) = (1-p)p …

I am little curious and can not figure out that how(why) is P(GB) equal to P(G)P(B) ?

In the given example, we are considering the scenario where we observe the gender of newborn children until a boy (B) is born. Let's break down the calculation of P(GB) step by step.

P(GB) represents the probability of observing a girl (G) followed by a boy (B). We can calculate this probability by considering the probabilities of each event happening in sequence.

First, we have P(G), which is the probability of observing a girl. In this context, since every birth is independent and the gender of each child is determined randomly, the probability of a child being a girl is equal to 1/2 or .5. Therefore, P(G) = .5.

Second, we have P(B), which is the probability of observing a boy. In this case, we are given p = P(B), which represents the probability of a single birth resulting in a boy. So, P(B) = p.

To calculate P(GB), we multiply the probabilities of each independent event happening in sequence. Since the births in this scenario are independent, we can multiply the probabilities as follows:

P(GB) = P(G) * P(B) = .5 * p = .5p.

Therefore, P(GB) is equal to P(G) multiplied by P(B), which gives us (1/2)p or .5p.

I hope this explanation clarifies why P(GB) is equal to P(G) * P(B). Feel free to ask if you have any further questions!