info2 wrote:Hi Mitch
I think now i understand the 4!/2!2! part.
3/6. 2/5. 3/4.2/3 here order within the boys has already been accounted for . same thing with the 2 girls group. When we multiply by 4! it will duplicate groups of boys and girls which have already been considered in the 3/6. 2/5. 3/4.2/3 part.
e.g B1B2GiG2 would be repeated when we multiply by 4! so to eliminate those repeated we need to divide by 2! twice.
Is this reasoning correct?
Thanks
Let's examine all of the ways to select exactly 2 boys and exactly 2 girls.
Case 1: Probability that the first and second selections are boys and the third and fourth selections are girls
P(BBGG) = 3/6 * 2/5 * 3/4 * 2/3 = 1/10.
Case 2: Probability that the first and second selections are boys and the second and fourth selections are girls
P(BGBG) = 3/6 * 3/5 * 2/4 * 2/3 = 1/10.
Case 3: Probability that the first and fourth selections are boys and the second and third selections are girls
P(BGGB) = 3/6 * 3/5 * 2/4 * 2/4 = 1/10.
Case 4: Probability that the second and third selections are boys and the first and fourth selections are girls
P(GBBG) = 3/6 * 3/5 * 2/4 * 2/3 = 1/10.
Case 5: Probability that the second and fourth selections are boys and the first and third selections are girls
P(GBGB) = 3/6 * 3/5 * 2/4 * 2/3 = 1/10.
Case 6: Probability that the third and fourth selections are boys and the first and second selections are girls
P(GGBB) = 3/6 * 2/5 * 3/4 * 2/3 = 1/10.
Every case above represents a different way to select exactly two boys and exactly two girls.
Thus, the probability of selecting exactly 2 boys and exactly 2 girls is equal to the SUM of the fractions above:
1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 = 1/10 * 6 = 3/5.
Notice the following:
The probability in each case is THE SAME (1/10).
The number of different cases -- 6 -- is equal to the number of ways to arrange the letters BBGG:
4!/2!2! = 6.
This is why we multiply the probability of Case 1 (3/6 * 2/5 * 3/4 * 2/3) by the number of ways to arrange the letters BBGG (4!/2!2!):
(3/6 * 2/5 * 3/4 * 2/3) * (4!/2!2!) = 1/10 * 6 = 3/5.
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