On an alien planet in which Zorgs (the aliens) only want Zags (one gender), every family continues to have children until they have a Zag. If they have a Zig (another gender), they have another child. If they have a Zag, they stop. What is the proportion of Zags to Zigs on the planet? What would cause your answer to be incorrect
Answers
Given : On an alien planet in which Zorgs (the aliens) only want Zags (one gender), every family continues to have children until they have a Zag
To find : proportion of Zags to Zigs on the planet?
Solution:
Probability of Zag = Zig = 1/2
Zags Zig Probability
1 0 1/2
1 1 1/4
1 2 1/8
1 3 1/16
1 4 1/32
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Mean of Zags = 1*1/2 + 1*1/4 + 1*1/8 + 1*1/16 + 1* 1/32 +................
= 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...............
a = 1/2 r = 1/2
= (1/2)/(1 - 1/2)
= 1
Mean of Zig = 0 * 1/2 + 1 * 1/4 + 2 * 1/8 + 3*1/16 + 4*1/32
= 1/4 + 2/8 + 3/16 + 4/32 +.........
Let say Mean of Zig = S
S = 1/4 + 2/8 + 3/16 + 4/32 +................
S/2 = 1/8 + 2/16 + 3/32 + 4/64 +............................
S - S/2 = 1/4 + (2/8 - 1/8) + (3/16 - 2/16 ) + ( 4/32 - 3/32) +.........................
S/2 = 1/4 + 1/8 + 1/16 + 1/32 + 1/64 +...................
a = 1/4 r = 1/2
=> S/2 = (1/4)/(1 - 1/2)
=> S/2 = 1/2
=> S = 1
Mean of Zig = 1
Zags : Zigs = 1 : 1
proportion of Zags to Zigs on the planet = 1:1
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