Probability of at Least One Girl in a Five-Child Family | Independent Births

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SUMMARY

The probability of having at least one girl in a family with five children, given that there is at least one boy, is calculated using conditional probability. The correct answer is 30/31, which contrasts with the initial incorrect calculation of 15/16. The solution involves understanding the binomial distribution and recognizing valid gender combinations that meet the problem's conditions. The key takeaway is the importance of applying conditional probability principles to solve such problems accurately.

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  • Understanding of binomial distribution
  • Knowledge of conditional probability
  • Familiarity with independent events in probability
  • Basic combinatorial analysis of outcomes
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  • Study the binomial distribution and its applications in probability
  • Learn about conditional probability formulas and their derivations
  • Explore examples of independent events in probability theory
  • Practice combinatorial problems involving gender distributions
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Students and educators in mathematics, particularly those focusing on probability theory, as well as anyone interested in understanding conditional probability and its applications in real-world scenarios.

tramp
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Hi
I have some difficulties with following question.

A family has five children. Assuming that the probability of a girl on each birth was 1/2 and that the five births were independent, what is the probability the family has at least one girl, given they have at least one boy?

My solution is 1-(1/2)^4 = 15/16

However according to the book correct answer it 30/31.
Any ideas?
 
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tramp said:
Hi
I have some difficulties with following question.

A family has five children. Assuming that the probability of a girl on each birth was 1/2 and that the five births were independent, what is the probability the family has at least one girl, given they have at least one boy?

My solution is 1-(1/2)^4 = 15/16

However according to the book correct answer it 30/31.
Any ideas?

Look at material on the binomial distribution.

The conditions in the problem are satisified if the family has any of the following combinations of genders:
(exactly 1 boy and 4 girls)
(exactly 2 boys and 3 girls)
(exactly 3 boys and 2 girls)
(exactly 4 boys and 1 girl)

The conditions are not satisfied by the combinations of genders:
(exactly 5 boys and 0 girls )
(exactly 0 boys and 5 girls )

The problem says we are "given" that the family has at least one boy, so you should look at the formula for conditional probability.
 
Thanks a lot Stephen.
I figured it out. Your post was a great help.
 

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