SOLUTION: Assume that the probability of the birth of a child of a particular sex is 50%. In a family with 4 children, what is the probability that: a). all children are boys b). all child

Question 1026251: Assume that the probability of the birth of a child of a particular sex is 50%. In a family with 4 children, what is the probability that:
a). all children are boys
b). all children are the same sex
c).there is at least one boy
I tried dividing out 50%, but I'm just more confused.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Let's list out the sample space. This is the entire set of all possibilities

Group 1:
BBBB

Group 2:
BBBG
BBGB
BGBB
GBBB

Group 3:
BBGG
BGBG
BGGB
GBGB
GGBB
GBBG

Group 4:
GGGB
GGBG
GBGG
BGGG

Group 5:
GGGG

B represents a boy. G represents a girl.
Group 1 shows that there is only one way to have all 4 boys and no girls.
Group 2 shows there are 4 ways to have three boys and one girl.
Group 3 has 6 possibilities of having 2 boys and 2 girls.
Group 4 is the opposite of group 2. There are 4 ways to have 3 girls and 1 boy
Group 5 is the opposite of group 1. There is only one way to have 4 girls and no boys

In total, there are 1+4+6+4+1 = 16 ways to have 4 kids

Things to note:
a) There are 4 slots and 2 choices per slot. It's no coincidence that 2^4 = 2*2*2*2 = 16
b) The sequence 1,4,6,4,1 is found in pascals triangle

Now onto the problem

----------------------------------------------------------------

a). all children are boys

As shown in the sample space above, there is only one way to have all 4 boys which is BBBB. The order of the boys doesn't really matter. This is out of 16 possible ways to have 4 kids.

The probability is 1/16 = 0.0625 = 6.25%

-------------------------------------------

b). all children are the same sex

"all children are the same sex" is basically saying "all boys" OR "all girls" (pick one side only)

There are 2 ways to either have all 4 boys or have all 4 girls. This is out of 16, so 2/16 = 1/8 = 0.125 = 12.5% is the answer for part b

-------------------------------------------

c).there is at least one boy

There are 2 ways to compute this. We could follow

method 1: count the number of combinations with at least one B in them and we'd count out 15 of 16. So the probability is 15/16 = 0.9375 = 93.75%
OR
method 2: the probability of having all girls is 6.25% which is the same as the probability of having all boys. Subtract this from 100% to get 100% - 6.25% = 93.75%

No matter which method you ultimatley go with, the final answer is 93.75% which can be represented as the fraction 15/16

--------------------------------------------------------------

In summary, the answers are..

(a) 1/16 = 0.0625 = 6.25%
(b) 1/8 = 0.125 = 12.5%
(c) 15/16 = 0.9375 = 93.75%

I'm showing all 3 formats (fraction, decimal, percent) as the answer. Choose whichever answer format your teacher wants to use.
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