A newly married couple decides to keep having children until they have a girl. What is the probability that the couple will have at least 5 children?
This is Geometric distribution.
Let Y be r.v."No: of chldn before a girl"
P(Y=y)=pq^(y-1)...therefore
P(Y=5)=(0.5)(0.5)^(5-1)=0.03125.
To calculate the probability that the couple will have at least 5 children until they have a girl, we need to consider the probability of having a girl (p) and the probability of having a boy (q). In this case, since they keep having children until they have a girl, we can assume the probability of having a boy or a girl is 0.5 (assuming the likelihood of having a boy or girl is equal).
To find the probability of having at least 5 children until they have a girl, we need to calculate the probability of having 4 boys before having a girl plus the probability of having 5 boys before having a girl, and so on.
The probability of having exactly k boys before having a girl is given by: P(X=k) = (q^k)*(p), where q is the probability of having a boy, p is the probability of having a girl, and k is the number of boys before the girl.
Now, let's calculate the probability of having at least 5 children until they have a girl.
P(at least 5 children) = P(4 boys before a girl) + P(5 boys before a girl) + P(6 boys before a girl) + ...
Since the probability of having a boy or a girl is 0.5, q = 0.5 and p = 0.5.
P(at least 5 children) = (0.5^4)*(0.5) + (0.5^5)*(0.5) + (0.5^6)*(0.5) + ...
Notice that this forms a geometric series with a common ratio of q = 0.5. The sum of an infinite geometric series can be calculated using the formula S = a / (1 - r), where "a" is the first term and "r" is the common ratio (0 < r < 1).
In this case, the first term is (0.5^4)*(0.5) and the common ratio is 0.5.
So, the probability of having at least 5 children until they have a girl is:
P(at least 5 children) = [(0.5^4)*(0.5)] / (1 - 0.5)
Simplifying, we have:
P(at least 5 children) = (0.5^5) / 0.5
Using exponent rules, we get:
P(at least 5 children) = 0.5^4 = 0.0625
Therefore, the probability that the couple will have at least 5 children until they have a girl is 0.0625 or 6.25%.