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 of these scenarios, we need to use the concept of geometric probability. Geometric probability calculates the likelihood of an event occurring after a certain number of trials, assuming each trial has a constant probability of success.
In this case, the probability of having a girl (success) is 0.5, assuming that boy and girl babies are equally likely. The probability of having a boy (failure) is also 0.5.
a) To find the probability that it would take at least three births to obtain two girls, we can consider two scenarios: the first two children are boys (BB), or the first two children are a boy and a girl (BG).
1) BB scenario: The probability of having a boy is 0.5, so the probability of having two boys in a row is (0.5)^2. However, in this case, we need at least three births, so after the first two boys, we need one more birth to get the desired outcome. Hence, the probability for this scenario is (0.5)^2 * 0.5 = 0.25.
2) BG scenario: The probability of having a boy and a girl in that order is 0.5 * 0.5 = 0.25. Again, after having a boy and a girl, we need one more birth to get the desired outcome.
To obtain the probability that it would take at least three births to obtain two girls, we need to find the probability of either the BB scenario or the BG scenario occurring. These scenarios are mutually exclusive, so we can add the probabilities: 0.25 + 0.25 = 0.5.
Therefore, the probability that it would take at least three births to obtain two girls is 0.5.
b) To find the probability that it would take at least four births to obtain two girls, we can consider two additional scenarios: BGG and GGB.
1) BGG scenario: The probability of having a boy followed by two girls is (0.5)^3 = 0.125. After having one boy and two girls, we need one more birth to get the desired outcome.
2) GGB scenario: The probability of having a girl followed by two boys is also (0.5)^3 = 0.125. After having two boys and one girl, we need one more birth to get the desired outcome.
Again, these scenarios are mutually exclusive, so we can add their probabilities: 0.125 + 0.125 = 0.25.
Therefore, the probability that it would take at least four births to obtain two girls is 0.25.
c) To find the probability that it would take at least five births to obtain two girls, we consider two more scenarios: GBGG and GGGB.
1) GBGG scenario: The probability of having a girl followed by a boy, then two girls, is (0.5)^4 = 0.0625. After having one girl, one boy, and two girls, we still need one more birth to get the desired outcome.
2) GGGB scenario: The probability of having three girls followed by a boy is also (0.5)^4 = 0.0625. Similarly, after having three girls and one boy, we need one more birth.
Adding the probabilities of these scenarios: 0.0625 + 0.0625 = 0.125.
Therefore, the probability that it would take at least five births to obtain two girls is 0.125.