Assuming boy and girl babies are equally likely, find the probability that it would take a) at least three births to obtain two girls b) at least four births to obtain two girls c) at least five births to obtain two girls
To find the probability in each scenario, we need to consider the possible outcomes and calculate their probabilities.
Let's start by understanding the scenarios:
a) "At least three births to obtain two girls" means that we are looking for the probability of having two girls after three or more births.
b) "At least four births to obtain two girls" means that we are looking for the probability of having two girls after four or more births.
c) "At least five births to obtain two girls" means that we are looking for the probability of having two girls after five or more births.
Now, let's break down each scenario step by step:
a) At least three births to obtain two girls:
To calculate this probability, we need to consider the possible outcomes of each birth sequence. We can have the following sequences of "B" (boy) and "G" (girl):
1) BBG
2) BGB
3) GBB
4) GGB
5) GBG
6) BGG
7) GGG
Out of these seven outcomes, three of them (GGG, GGB, GBG) meet the condition of having at least two girls. Therefore, the probability is 3/7.
b) At least four births to obtain two girls:
In this case, we need to consider all possible birth sequences that meet the condition of having at least two girls. Again, we have the following sequences:
1) BBGG
2) BGGB
3) BGBG
4) GBGB
5) GBBG
6) BGGG
7) GBGG
8) GGBG
9) GGGB
Out of these nine outcomes, six of them (GGGB, GGGB, GGBG, GBGG, BGGG, GGGG) meet the condition of having at least two girls. Therefore, the probability is 6/9 or 2/3.
c) At least five births to obtain two girls:
Similarly, we need to consider all possible birth sequences that meet the condition of having at least two girls:
1) BBGGG
2) BGBGG
3) BGGBG
4) BGGGB
5) GGBGG
6) GBGBG
7) GBGGB
8) GGGBG
9) GGGGB
Out of these nine outcomes, four of them (GGGGB, GGGBG, GGBGG, GBGGG) meet the condition of having at least two girls. Therefore, the probability is 4/9.
So, to summarize:
a) The probability of at least three births to obtain two girls is 3/7.
b) The probability of at least four births to obtain two girls is 2/3.
c) The probability of at least five births to obtain two girls is 4/9.