A statistician believes that in her town each child born has a 50% chance of being a girl, independently of all other children. There are 281 three-child families in her town; 39 of these families have no girls, 94 have one girl, 115 have two girls, and the remaining 33 have three girls.
1) If the statistician’s belief is correct, the probability that a three-child family in the town will have exactly one girl is ______%.
2) If the statistician’s belief is correct, then among the three-child families in the town what is the expected number of families that have exactly one girl?
3) If the statistician’s belief is correct, then among the three-child families in the town what is the expected number of families that have no girls?
4) The P-value of the appropriate test is about ________%.
1) To calculate the probability that a three-child family will have exactly one girl, we need to use the concept of binomial probability.
The probability of an individual child being a girl is 0.50 since each child has a 50% chance of being a girl.
Now, we can use the formula for binomial probability:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
where:
P(X = k) is the probability of getting exactly k successes (in this case, one girl),
n is the total number of trials (in this case, 3 children),
p is the probability of success (here, the probability of having a girl in a single birth event),
k is the number of successful trials (here, one girl).
Plugging in the values, we have:
P(X = 1) = (3C1) * (0.50)^1 * (1-0.50)^(3-1)
Using the combination formula (nCr) to calculate (3C1) = 3, we can calculate the probability:
P(X = 1) = 3 * 0.50 * (1-0.50)^2 = 3 * 0.50 * 0.25 = 0.375 or 37.5%
Therefore, the probability that a three-child family in the town will have exactly one girl is 37.5%.
2) To find the expected number of families that have exactly one girl, we can multiply the probability we calculated in the previous step by the total number of families in the town.
Expected number of families = Probability * Total number of families = 0.375 * 281 = 105.375
Therefore, the expected number of families that have exactly one girl is approximately 105.
3) Similarly, to find the expected number of families that have no girls, we can use the same approach.
The probability of having no girls in a single birth event is (1-0.50) = 0.50.
Using the formula for binomial probability, we have:
P(X = 0) = (3C0) * (0.50)^0 * (1-0.50)^(3-0) = 1 * 1 * 0.125 = 0.125 or 12.5%.
Expected number of families = Probability * Total number of families = 0.125 * 281 = 35.125
Therefore, the expected number of families that have no girls is approximately 35.