Kushchokhani wrote:
A couple decides to have 5 children. Each child is equally likely to be a boy or a girl. What is the probability that the couple has exactly 3 girls and 2 boys?
A. 1/32
B. 1/16
C. 5/32
D. 5/16
E. 1/6
I. Working on the number of cases\(12345\)
a) Say 1 is G, other 2 Gs can be any out of 2, 3, 4 or 5. => 1*4C2=6
b) Say 1 is not G but 2 is G, other 2 Gs can be any out of 3, 4 or 5. => 1*1*3C2=3
c) Say 1 and 2 are not Gs but 3 is G, other 2 Gs can be any out of 4 or 5. => 1*1*1*2C2=1
Total favorable ways = 6+3+1=10
Total ways = 2*2*2*2*2=32
P = \(\frac{10}{32}=\frac{5}{16}\)
II. Direct calculation of P\(5C3*(\frac{1}{2})^3*(\frac{1}{2})^2=\frac{5}{16}\)
Himaanshu007, the reason why arrangement is important..
If P of 3 G and 2 B is 1/32, then following cases will also have P as 1/32
(1G,4B); (2G,3B); (3G,2B); (4G,1B); (5G,0B) and (0g,5B)............6cases, so combined P = 6*1/32=6/32.
But there is NO other possibility apart from 6 cases, and so they should add up to 1. where do \(1-\frac{6}{32}\) or 26/32 go?
When you calculate P as 1/2 *1/2....., what is 2 in the denominator? OR when you calculate total ways as 2*2*2*2*2 or 32, what does 2 stand forIt is for B and G, so B*B*G*G*G is different from G*B*B*B*B.