cashier ticketing problem: A cashier has a total of 400 tickets of five different colours Blue, Green, Red, Yellow and Orange.

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nanase

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A cashier has a total of 400 tickets of five different colours Blue, Green, Red, Yellow and Orange. The ratio of blue to green to red is 1 : 2 : 4. The ratio of green to yellow to orange tickets is 1 : 3 : 6. What is the smallest number of tickets that must be drawn to ensure that at least 50 tickets of one colour have been selected?

Please help me out with the steps.
I tried forming the ratio for all colours and then obtain roughly (not exact) 105 tickets?
I am not quite sure how to interpret the last sentence though, does it mean how many tickets I should take so that I will ensure 50 tickets of the same one colour?
What I have done is I look at orange (which has the biggest ratio). The total of the ratio numbers is 25 and I will have 12 orange tickets, so to make sure I have 50 orange tickets, I should draw 105 tickets?
tickets ans.jpeg
 
nanaseailie said:
A cashier has a total of 400 tickets of five different colours Blue, Green, Red, Yellow and Orange. The ratio of blue to green to red is 1 : 2 : 4. The ratio of green to yellow to orange tickets is 1 : 3 : 6. What is the smallest number of tickets that must be drawn to ensure that at least 50 tickets of one colour have been selected?

Please help me out with the steps.
I tried forming the ratio for all colours and then obtain roughly (not exact) 105 tickets?
I am not quite sure how to interpret the last sentence though, does it mean how many tickets I should take so that I will ensure 50 tickets of the same one colour?
What I have done is I look at orange (which has the biggest ratio). The total of the ratio numbers is 25 and I will have 12 orange tickets, so to make sure I have 50 orange tickets, I should draw 105 tickets?
View attachment 36741
Click to expand...
First, determine the number of tickets of each color then think about the worst possible outcome that you could have drawn the tickets.
 
because there are 400 tickets
Blue 16
green 32
red 64
yellow 96
orange 192
my logic is since orange is the biggest, I should focus on getting 50 of that colour.
[math]\frac{192}{400}=\frac{50}{x}[/math]still getting x =104.166667 or 105 tickets that needs to be drawn to ensure getting 50 orange.
Is this right? @BigBeachBanana
 
Last edited:
nanaseailie said:
because there are 400 tickets
Blue 16
green 32
red 64
yellow 96
orange 192
my logic is since orange is the biggest, I should focus on getting 50 of that colour.
[math]\frac{192}{400}=\frac{50}{x}[/math]still getting x =104.166667 or 105 tickets that needs to be drawn to ensure getting 50 orange.
Is this right? @BigBeachBanana
Click to expand...
No, it's not: you can get 49 orange, 49 yellow and 7 red, just to pick one example.
Rephrasing @BigBeachBanana's advice: how many tickets can you pick without getting 50 tickets of the same color? What is the maximum number of tickets after which you cannot avoid having 50 of the same color?
 
blamocur said:
What is the maximum number of tickets after which you cannot avoid having 50 of the same color? What is the maximum number of tickets after which you cannot avoid having 50 of the same color?
Click to expand...
Ahhh, nice thank you! This line got me thinking differently.
So if I take
Blue 16
green 32
red 49
yellow 49
orange 49
total of 195 tickets and I can avoid having 50 of the same colour. If I take 196 tickets I will ensure either red/yellow/orange will be 50.
Just curious for my development, is there like a smarter mathematical way for this? just curious.

Thank you so much @blamocur and @BigBeachBanana
 
Last edited:
nanaseailie said:
for my development...
Click to expand...
You can attempt to generalize the problem.

Given [imath]n[/imath] categories (e.g. blue, green, red...), and their corresponding counts [imath]k_n[/imath]. What is the smallest number of draws to ensure at least [imath]m[/imath] counts, where [imath]m \le \sum_{i=1}^nk_i[/imath], of one category have been selected?
 
BigBeachBanana said:
You can attempt to generalize the problem.

Given [imath]n[/imath] categories (e.g. blue, green, red...), and their corresponding counts [imath]k_n[/imath]. What is the smallest number of draws to ensure at least [imath]m[/imath] counts, where [imath]m \le \sum_{i=1}^nk_i[/imath], of one category have been selected?
Click to expand...
Wouldn't you want to have [imath]m\leq \max_i k_i[/imath] ?
 
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