1. In families with four children, you’re interested in the probabilities for th

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

Explanation / Answer

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) = nCx Px 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) = nCx Px 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) = nCx Px 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) = nCx Px 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) = nCx Px 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

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

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

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