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S2 Poisson, answer not working out same as book watch

1. I want to check that my working is correct, book answer is different from mine.

Question. At the hot drinks counter in a cafeteria both tea and coffee are sold. The number of cups of coffee sold per minute may be assumed to be a Poisson variable with mean 1.5 and the number of cups of tea sold per minute may be assumed to be an independent Poison variable with mean 0.5.

My answer for the last part of the problem is not the same as the book answer. I am wondering if book is incorrect or me.

(iii) In a given one-minute period exactly three drinks are sold. Calculate the probability that these are all cups of coffee.

I approached it like this.

C ~ Poisson(1.5) and T ~ Poisson(0.5)

for coffee
P(X = 3) = 0.1255

for tea
P(X = 0) = 0.6065

[I double checked P values]

P(3 coffees, zero teas) = 0.1255 x 0.6065 = 0.07612

But book says 0.422.

What have I done wrong???
2. The number of drinks sold in a minute can be expressed as D~Poisson(1.5+0.5) i.e. D~Poisson(2). We can do this because the number of coffees sold and number of teas sold are independent.

The question is asking you for Pr(C=3 and T=0 | D=3).

Normally you would do Pr(A=a|B=b)=Pr(A=a and B=b)/Pr(B=b). Can you use this and apply it to the above to get the right answer?
You'll need to use that if A and B are independent, Pr(A=a and B=b)=Pr(A=a)*Pr(B=b).
3. (Original post by rayquaza17)
The number of drinks sold in a minute can be expressed as D~Poisson(1.5+0.5) i.e. D~Poisson(2). We can do this because the number of coffees sold and number of teas sold are independent.

The question is asking you for Pr(C=3 and T=0 | D=3).

Normally you would do Pr(A=a|B=b)=Pr(A=a and B=b)/Pr(B=b). Can you use this and apply it to the above to get the right answer?
You'll need to use that if A and B are independent, Pr(A=a and B=b)=Pr(A=a)*Pr(B=b).
Hi, that's back to S1 material - it's been a long time.

P(D=3) = 0.1804

P(T=0) = 0.6065

P(C=3) = 0.1255

Sp P(coffee=3 and tea=0) =

P(C=3) x P(T=0)
------------------------
P(D=3)

= 0.1255 x 0.6065
-------------------------
0.1804

= 0.422 which is correct book answer :-)

Thank you.

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Updated: January 11, 2015
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