A family has two children. What is the probability that both children are boys given that at least one of them is a boy?
Understand the Problem
The question involves two parts: finding the median of a given distribution and calculating the probability of a family having two boys given that at least one child is a boy. These require statistical analysis and basic probability concepts.
Answer
Median = 18, Probability = $\frac{1}{3}$.
Answer for screen readers
The median of the distribution is 18, and the probability that both children are boys, given at least one is a boy, is $\frac{1}{3}$.
Steps to Solve
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Identify the total values and frequencies From the distribution table, you can extract the values of $x$ and their corresponding frequencies $f$:
- $x$: 11, 14, 12, 18, 13, 15, 19, 17, 16
- $f$: 4, 16, 7, 20, 10, 18, 18, 22, 25
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Calculate the cumulative frequency You need to find the cumulative frequency for each $x$ value by adding the frequencies together up to that point.
- For example:
- Cumulative frequency for 11: 4
- Cumulative frequency for 14: $4 + 16 = 20$
- Cumulative frequency for 12: $20 + 7 = 27$, and so on.
- For example:
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Find the total number of observations Add all the frequencies to get the total number of observations: $$ N = 4 + 16 + 7 + 20 + 10 + 18 + 18 + 22 + 25 = 120 $$
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Determine the median position To find the median, use the formula: $$ \text{Median position} = \frac{N}{2} = \frac{120}{2} = 60 $$ This means the median will be the value of $x$ at the 60th observation.
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Locate median class Use the cumulative frequencies to find where the 60th observation falls. Identify the smallest cumulative frequency that is greater than or equal to 60.
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Find median value The median value will correspond to the class where the cumulative frequency first exceeds 60. Use interpolation if necessary.
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Conditional probability setup For the second part of the question, consider the possible outcomes for two children:
- BB (both boys)
- BG (boy and girl)
- GB (girl and boy)
- GG (both girls)
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Count relevant outcomes Given that at least one child is a boy, the possible outcomes are:
- BB, BG, GB (3 outcomes)
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Calculate desired probability The probability that both children are boys given that at least one is a boy is: $$ P(BB | \text{at least one boy}) = \frac{P(BB)}{P(BB) + P(BG) + P(GB)} = \frac{1}{3} $$
The median of the distribution is 18, and the probability that both children are boys, given at least one is a boy, is $\frac{1}{3}$.
More Information
The median is a measure that indicates the middle value of a dataset. When calculating conditional probabilities, it's essential to clearly define the sample space and favorable outcomes.
Tips
- Confusing the cumulative frequency with just the frequency of values. Ensure to sum frequencies correctly.
- Forgetting that conditional probabilities require the adjustment of the sample space based on given information. This leads to incorrect calculations.
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