What is the probability that 5 large loaves are heavier than 10 small loaves?

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SUMMARY

The probability that 5 large loaves of bread are heavier than 10 small loaves can be calculated using the properties of normal distributions. The weight of a large loaf follows a normal distribution with mean 420g and standard deviation 30g, while a small loaf has a mean of 220g and standard deviation of 10g. The correct approach involves recognizing that the total weight of 5 large loaves is normally distributed as N(2100, 22500), while the total weight of 10 small loaves is N(2200, 1000). The final probability calculation shows that the weights are not directly comparable as initially assumed.

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The weight of a large loaf of bread is a normal variable with mean 420g and standard deviation 30g. The weight of a small loaf is a normal variable with mean 220g and standard deviation 10g.

1) Find the probability that 5 large loaves of bread are heavier than 10 small loaves.

My Working:
Let X be the weight of a large loaf, Y be the weight of a small loaf

X~N(420, 900) , Y~N(220,100)

5X>10Y
5X-10Y>0
X-2Y>0

E(X-2Y)=E(X) - 2E(Y)=420-440=-20
Var(X-2Y)=Var(X)+4Var(Y)=900+400=1300

P(X-2Y>0)=P(Z>20/(1300)^1/2)=P(Z>0.5547)=0.2896

But the answer given is totally different. Is there anything I miss in my working?
 
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Are X,Y normally distributed random variables? If so, then:

Z_{X} = \frac{X-420}{30}, and Z_{Y} = \frac{Y-220}{10}

Also var(aX+bY) = a^{2}var(X) + b^{2}var(Y) + 2abcov(X,Y)

Since X,Y are independent, then cov(X,Y) = 0
 
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You are working with different loaves of bread here, with EACH of their weights being normally distributed. It would not be correct to consider the case where 5X > 10Y, because this would imply you are thinking about the scenario where 5 times the weight of ONE large loaf is greater than 10 times the weight of ONE small loaf.

Instead you should think about the distribution of X_{1} + X_{2} + X_{3} + X_{4} + X_{5}.

Observe that 5X \sim N(2100,22500) but X_{1} + X_{2} + X_{3} + X_{4} + X_{5} \sim N(2100,4500), so these 2 distributions are indeed different.
 
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Thanks, I understand it now.