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Chapter 7: Problem 31

In a three-child family, what is the probability that all three children are girls given that at least one of the children is a girl? (Assume that the probability of a boy being born is the same as the probability of a girl being born.)

Short Answer

Expert verified
The probability of having all three children be girls given that at least one of them is a girl is \(\boxed{\frac{1}{7}}\).

Step by step solution

01

Find the probability of GGG

Each child can be either a boy (B) or a girl (G) with a probability of 1/2 for each. The probability of having all girls (GGG) is the product of three individual probabilities: \(P(GGG) = P(G) * P(G) * P(G)\)
02

List out possibilities of having at least one girl

Any of the three children being a girl satisfies the condition of having at least one girl. We can represent all possible combinations of having at least one girl with these combinations: GGG (all girls) GGB (first and a second child are girls) GBG (first and a third child are girls) BGG (second and the third child are girls) GBB (only the first child is a girl) BGGB (only the second child is a girl) BBG (only the third child is a girl) Note that the only missing combination is BBB, which is the case of three boys.
03

Find the probability of having at least one girl

Now, to find the probability of having at least one girl, we add the probabilities of each combination: - GGG: \(1/2 * 1/2 * 1/2 = 1/8\) - GGB: \(1/2 * 1/2 * 1/2 = 1/8\) - GBG: \(1/2 * 1/2 * 1/2 = 1/8\) - BGG: \(1/2 * 1/2 * 1/2 = 1/8\) - GBB: \(1/2 * 1/2 * 1/2 = 1/8\) - BGGB: \(1/2 * 1/2 * 1/2 = 1/8\) - BBG: \(1/2 * 1/2 * 1/2 = 1/8\) Adding these probabilities, we get \(P(\text{at least one girl}) = 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 = 7/8\)
04

Apply conditional probability formula

We want to find the probability of all three children being girls (GGG) given that at least one of them is a girl. Therefore, we will use the conditional probability formula: \[P(GGG|\text{at least one girl}) = \frac{P(GGG \cap \text{at least one girl})}{P(\text{at least one girl})}\] Since the event 'GGG' is a subset of the event 'at least one girl. The probability of GGG and at least one girl is equal to the probability of GGG. So, \[P(GGG|\text{at least one girl}) = \frac{P(GGG)}{P(\text{at least one girl})} = \frac{1/8}{7/8}\] \[P(GGG|\text{at least one girl}) = \frac{1}{7}\] So, the probability of having all three children be girls given that at least one of them is a girl is \(\boxed{\frac{1}{7}}\).

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