Discrete Random Variable Problem

[dreamingotter]

There are 3 ice cream flavors: vanilla, chocolate, and strawberry, each with probabilities 0.3, 0.5, and 0.2
Given that out of the last 120 customers, 30 ordered vanilla, I want to know the distribution of how many chocolates were among the 120 orders.

Would it be incorrect to think about this as a binomial distribution with n=90 and p=0.5? I'm not too sure how to approach this problem...

 

[Steven G]

You want to know how many of the 120 ice creams sold were chocolate? How can you predict how many chocolate ice creams were sold? If you were also told some strawberry were order than that is another story. Is this the entire problem?

 

[Romsek]

\(\displaystyle \text{Out of the remaining 90 cones we will have}\\
p[\text{chocolate}] =\dfrac 5 7,~ p[\text{strawberry}]=\dfrac 2 7\\
p[k \text{ chocolate}] = \dbinom{90}{k} \left(\dfrac 5 7\right)^k \left(\dfrac 2 7\right)^{90-k}
\)

so you were pretty close but you forgot to renormalize the probabilities of chocolate and strawberry.

 

[dreamingotter]

\(\displaystyle \text{Out of the remaining 90 cones we will have}\\
p[\text{chocolate}] =\dfrac 5 7,~ p[\text{strawberry}]=\dfrac 2 7\\
p[k \text{ chocolate}] = \dbinom{90}{k} \left(\dfrac 5 7\right)^k \left(\dfrac 2 7\right)^{90-k}
\)

so you were pretty close but you forgot to renormalize the probabilities of chocolate and strawberry.

That makes sense. Thank you so much!

 

[dreamingotter]

You want to know how many of the 120 ice creams sold were chocolate? How can you predict how many chocolate ice creams were sold? If you were also told some strawberry were order than that is another story. Is this the entire problem?

Yes, this is the entire problem. Romsek helped me out with this one. Thanks though!

 

[Steven G]

Professor, I do see 100% how you got p[chocolate] = 5 /7, and p[strawberry]= 2/7 but that does not mean I agree with (but am sure you are correct)

I suspect that maybe I am not understanding the question. Especially since p(vanilla) did not live up to being .3, how can you say that the remaining flavors will live up to their given probability?
Thanks!