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In a family having three children, there may be no girl, one girl, two girls, or three girls. So, the probability of each is 1/4. Is this correct? Justify your answer.

User Muirbot
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Final answer:

No, the probability of each scenario (no girl, one girl, two girls, or three girls) in a three-child family is not 1/4. The correct probabilities are 1/8 for no girls or three girls, and 3/8 for one girl or two girls, assuming equal chances for each sex.

Step-by-step explanation:

The probability of having no girl, one girl, two girls, or three girls in a family with three children is not equally likely and therefore not 1/4 for each scenario. This assumption would only be correct if each outcome had an equal chance of occurring. However, the genders of the children born to a couple can be modeled as independent events, with each event (birth of a child) having two possible outcomes: boy or girl.

Assuming the probability of having a boy or girl is equal (which is slightly simplified as the actual ratio is approximately 105 boys to 100 girls), each child has a 1/2 chance of being a girl. The different scenarios for 0, 1, 2, or 3 girls can be calculated using the binomial distribution:

  • Zero girls (BBB): (1/2)^3 = 1/8
  • One girl (GBB, BGB, BBG): 3 * (1/2)^3 = 3/8
  • Two girls (GGB, GBG, BGG): 3 * (1/2)^3 = 3/8
  • Three girls (GGG): (1/2)^3 = 1/8

As you can see, the probability of having one or two girls is each 3/8, while the probability of no girls or three girls is each 1/8. These probabilities do not equate to 1/4 as initially stated.

User Slillibri
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