Question

OK, so you wrecked your mother's classic 1964 lipstick red GTO convertible with a 348hp Tri-powered 389, a whip antenna and those really neat fuzzy dice, which is why you just spent your last paycheck on scratch offs. Seems that Firestone wants $12,000 to fix the damage and your current net worth is $125.75 and a linty Altold. You are most assuredly DOOOOOMED. But you don't need to worry thanks to MODERN GENETICS. As luck would have it you're best friend happens to be a molecular geneticist who has developed a strain of chicken that carries the gene for GOLD EGGS! If a fertilized egg is homozygous for the gene "au" it will be SOLID GOLD, otherwise it's just a chick. Considering the current spike in the price of gold each x-large au/au egg is worth $1750. Your friend has promised you the next dozen eggs from a chicken he has hidden away, BUT there is no guarantee how many of the eggs will be gold. Considering these facts what is the probability that you will survive the WRATH OF MOM? n-X P= n! x!(n-x)! ip*q"* P.S. You might need this:

Answer 1

Cost of repairing the damage = $12000

Worth of 1 Gold Egg = $1750

Minimum number of Gold Eggs required = $$12000\div $1750 = 6.85\approx 7 \:golden\: eggs$

So, we need at least 7 golden eggs.

Consider 2 alleles involved in fertilisation to be GOLD(AU) and NOT GOLD (NAU).

So, the possible combinations of the alleles are ( AU, AU), (AU,NAU) , (NAU, AU), (NAU,NAU).

Each of these combinations have probability of 1/4.

For the fertilised egg to be homozygous, there must be same alleles for a particular gene from both parents.

So, the homozygous combinations are (AU, AU) and (NAU,NAU). But to obtain golden eggs we need the homozygous combination of Gold i.e (AU,AU).

Thus,

Probability of getting a golden egg

=

= 1/4

Thus ,
Probability of not getting a golden egg

Number of eggs given by your friend = 12

So, if we obtain 7 or more Golden Eggs , we will be saved.

Thus, we have to find the probability of getting at least 7 Golden Eggs out of 12 eggs.

So, number of Golden Eggs out of 12 eggs follow .

Binomial(12, 1/4)

Probability of getting Golden Eggs out of n Eggs  

T

n-2 n! pq x!(n - ..)!

In our case, n = 12, p = 1/4 and q = 3/4 and = 7,8,9,10,11,12

T

Probability of getting Golden Eggs out of 12 Eggs

T

  $=\frac{12!}{x!(12-x)!}\:\frac{1}{4}^{x} \frac{3}{4}^{12-x}$

Putting = 7,8,9,10,11 and 12 in the above formula,

T

$=\frac{12!}{7!(12-7)!}\:\frac{1}{4}^{7} \frac{3}{4}^{12-7}+\frac{12!}{8!(12-8)!}\:\frac{1}{4}^{8} \frac{3}{4}^{12-8}+\frac{12!}{9!(12-9)!}\:\frac{1}{4}^{9} \frac{3}{4}^{12-9}+\frac{12!}{10!(12-10)!}\:\frac{1}{4}^{10} \frac{3}{4}^{12-10}+\frac{12!}{11!(12-11)!}\:\frac{1}{4}^{11} \frac{3}{4}^{12-11}+\frac{12!}{12!(12-12)!}\:\frac{1}{4}^{12} \frac{3}{4}^{12-12}$

$= 0.01147+ 0.00238+0.000354+0.0000354+0.0000021+ 5.96\times 10^{-8}\\ = 0.01425278$

Thus, the probability of getting at least 7 golden eggs out of 12 eggs is 0.01425278.So, the probability that you will survive the wrath of your mom is 0.01425278.