# Ratio of Boys and Girls in a Country where people want only boys

**Total Probability** =ΣProbability of event = 1/2 +1/4 +1/8 + …**= 1**

Now,

**Expected no of Boys** will be

**= Σ Probability of event x Number of Boys in that event**

=1/2 x1 + 1/4 x1 + 1/8 x 1 + ….

= 1/2+1/4+1/8+1/16+…

**=** **1**

**Expected no of Girls** will be

**= Σ Probability of event x Number of Girls in that event**

=1/2 x0 + 1/4 x1 + 1/8 x 2 + 1/16 x 3 …. (See tree diagram)

= 1/4+ 2/8 + 3/16+ 4/32 …

=Sn

Sn = 1/4+ 2/8 + 3/16+ 4/32 +…

1/2x Sn = 1/8 + 2/16 + 3/32+ 4/64 +… (Shifting all terms to right)

————————————————————-

Sn/2 = 1/4+ 1/8+ 1/16 + 1/32 +… (Subtracting)

Sn/2 = 1/4 x 2

Sn = 1/4 x 2 x 2 =1

**Expected no of Girls will be =1 **

Therefore

**Ratio of no of Boys to Girls =**

__Expected no of Boys __** = **__1__** = 1:1**

**Expected no of Girls 1**

```
Answer: The ratio is 1:1 (surprisingly)
Explanation: This is a simple question of probability...
Suppose we have C families. So we will eventually have C sons (after all, we keep giving birth until a son is born)
The probability of having a daughter is:
Number of girls = 0 * the probability of having 0 girls + 1 * the probability of having one girl + 2 * the probability of having 2 girls + ......
We will get: 0*(C*1/2) + 1*(C*1/2*1/2) + 2*(C*1/2*1/2*1/2) + …
That is: 0 + C/4 + 2*C/8 + 3*C/16 + …
It converges to C. So the ratio is 1:1
```

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