1. ## Probability

A family with 5 children is selected at random.What is the probability that the family has at
least 3 boys?

2. Hello, Jhonson!

A family with 5 children is selected at random.
What is the probability that the family has at least 3 boys?
Here's an "eyeball" solution . . .

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: .$\displaystyle P(\text{3 or more boys}) \:=\:\frac{1}{2}$

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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: .$\displaystyle P(\text{3 or more boys}) \:=\:\frac{16}{32} \:=\:\frac{1}{2}$