A family with 5 children is selected at random.What is the probability that the family has at
least 3 boys?
Hello, Jhonson!
Here's an "eyeball" solution . . .A family with 5 children is selected at random.
What is the probability that the family has at least 3 boys?
Of all the possible combinations of gender among five children,
. . exactly HALF of them have more boys than girls.
. . (And the other half has more girls than boys.)
Therefore: .$\displaystyle P(\text{3 or more boys}) \:=\:\frac{1}{2}$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
A more "mathematical" solution . . .
There are: .$\displaystyle 2^5 = 32$ possible sets of gender.
How many have 3 boys and 2 girls?. . $\displaystyle _5C_3 = 10$
How many have 4 boys and 1 girl?. . $\displaystyle _5C_4 = 5$
How many have 5 boys?. . $\displaystyle _5C_5 = 1$
Hence, there are: .$\displaystyle 10 + 5 + 1 \:=\:16$ cases with 3 or more boys.
Therefore: .$\displaystyle P(\text{3 or more boys}) \:=\:\frac{16}{32} \:=\:\frac{1}{2}$