Question

1. In families with four children, you’re interested in the probabilities for the different possible numbers of girls in a family. Using theoretical probability (assume girls and boys are equally likely), compile a five-column table with the headings “0” through “4,” for the five possible numbers of girl children in a four-child family. Then, using “G” for girls and “B” for boys, list under each heading the various birth-order ways of achieving that number of girls in a family.

Then, use your table to calculate the following probabilities:

a. The probability of 1 girl
b. The probability of 2 girls
c. The probability of 4 girls
d. The probability the third child born is a girl

Answer 1

Solution:-

There are 50% chances that selected child would be girl or boy (assume girls and boys are equally likely)

Hence P :- Probability of success (selected child is a girl) = 0.5

Q :- Probability of failure (selected child is a boy) = 0.5

using Binomial distribution

P(X=x) = ⁿC_x P^x Q^(n-x)

P(X = 0) i.e No girl is selected

P(X = 0 ) = ⁴C₀ * (0.5)⁰ * (0.5)^(4-0)

P(X = 0 ) = 1 * 1* 0.0625 = 0.0625

P(X = 1) i.e 1 girl is selected

P(X = 1) = P(X=x) = ⁿC_x P^x Q^(n-x)

P(X = 1 ) = ⁴C₁* (0.5)¹ * (0.5)^(4-1)

P(X = 1 ) = 4 * 0.5 * 0.125 = 0.25

P(X = 2) i.e 2 girl is selected

P(X = 2) = P(X=x) = ⁿC_x P^x Q^(n-x)

P(X = 2 ) = ⁴C₂* (0.5)² * (0.5)^(4-2)

P(X = 2 ) = 6 * 0.25 * 0.25 = 0.375

P(X = 3) i.e 3 girl is selected

P(X = 3) = P(X=x) = ⁿC_x P^x Q^(n-x)

P(X = 3 ) = ⁴C₃* (0.5)³ * (0.5)^(4-3)

P(X = 3 ) = 4 * 0.125 * 0.5 = 0.25

P(X = 4) i.e 4 girl is selected

P(X = 4) = P(X=x) = ⁿC_x P^x Q^(n-x)

P(X = 4 ) = ⁴C₄* (0.5)⁴ * (0.5)^(4-4)

P(X = 4 ) = 1 * 0.0625 * 1 = 0.0625

Other Method is

Combination of boy and girl in 4 childern is 2⁴ = 16

so our sample space would be

BBBB BBBG BBGB BBGG BGBB BGBG BGGB BGGG GBBB GBBG GBGB GBGG GGBB GGBG GGGB GGGG Selecting 0 Girl = BBBB = 1/16 = 0.0625 Selection 1 Girl = {BBBG BBGB BGBB GBBB } = 4 = 4/16 = 0.25 Selecting 2 Girl = {BBGG BGBG BGGB GBBG GBGB GGBB} = 6 = 6/16 = 0.375 Selecting 3 Girl = {BGGG GBGG GGBG GGGB} = 4 = 4/16 = 0.25 Selecting 4 Girl = {GGGG} = 1 = 1/16 = 0.0625 Number of Girl Selected 0 1 2 3 4 Probability 0.0625 0.25 0.375 0.25 0.0625