A sample of four families with three children each was taken.find the probability that one family has two girls,two families have three girls and one family has three boys.
To find the probability of this specific combination, we need to calculate the total number of possible outcomes and the number of favorable outcomes.
Step 1: Calculate the total number of possible outcomes
In this case, each family has three children, and each child can be either a boy or a girl. So, for each family, the total number of possible outcomes is 2^3 = 8.
Since we have four families, the total number of possible outcomes for the four families is 8^4 = 4096.
Step 2: Calculate the number of favorable outcomes
We want one family to have two girls (GG), two families to have three girls (GGG), and one family to have three boys (BBB).
To calculate the number of favorable outcomes, we need to consider the different ways these combinations can occur across the four families.
For one family to have two girls (GG), we can choose this family in four different ways (family 1, family 2, family 3, or family 4).
For two families to have three girls (GGG), we can select these families in (4 choose 2) = 6 different ways.
Once we have selected the two families for three girls (GGG), we can arrange the three girls in each family in 2^3 = 8 different ways (each child can be either a girl or a boy).
For the remaining family to have three boys (BBB), there is only one possible arrangement.
Therefore, the number of favorable outcomes is 4 * 6 * (8 * 8 * 8) * 1 = 12288.
Step 3: Calculate the probability
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 12288 / 4096
Probability ≈ 3
Therefore, the probability that one family has two girls, two families have three girls, and one family has three boys is approximately 3/4096.