Problems in math of probabilities: 5 yellow socks, 6 red socks, 7 orange socks; magic mosquitoes

Ryan$

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Hi guys, I really found it hard to solve, so I need your help and illustrate to me please the steps to solve those two questions.

One question in probability:
there're Socks which they distributed like this: 5 yellow socks, 6 red socks, 7 orange socks.
how many socks we must take out in order to have two identical socks? (( two identical socks= two matching socks, in other words the same color .. ))

second question:
There is a mosquito created in time 0 , after two seconds he starts to generate at every second "two mosquito", what happened at second "8.5" (after 8 seconds)? (in other words how many mosquito we would have at after 8 seconds?)
my answer here, I used Fibonacci sequence but I guess my answer is wrong because I get answer 20 mosquito .. totally wrong!


please at your solution make sure to illustrate the steps of the solution, thanks alot !!
 

Harry_the_cat

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One question in probability:
there're Socks which they distributed like this: 5 yellow socks, 6 red socks, 7 orange socks.
how many socks we must take out in order to have two identical socks? (( two identical socks= two matching socks, in other words the same color .. ))
We must assume here that we are taking the socks out blindly.
If you take out three socks, you might get 1 yellow 1 red and 1 orange. If you take out 4 socks you must get at least one pair of identical socks. Think about it.
A follow up question: How many socks do you have to take out to have two yellow socks?
 

Ryan$

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We must assume here that we are taking the socks out blindly.
If you take out three socks, you might get 1 yellow 1 red and 1 orange. If you take out 4 socks you must get at least one pair of identical socks. Think about it.
A follow up question: How many socks do you have to take out to have two yellow socks?
7+6+2 = 15
 

Ryan$

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what about the second question of generating mosquito?! thanks alot.
 

Harry_the_cat

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what about the second question of generating mosquito?! thanks alot.
t=0 ... 1 mosquito
t=2 … original 1 mosquito + 2 mosquitoes = 3 mosquitoes
t=3 … 3 mosquitoes + 1x2 mosquitoes = 5 mosquitoes (the new two aren't ready to "reproduce" yet)
t=4 … 5 mosquitoes + 3x2 mosquitoes = 11 mosquitoes (the new two aren't read to "reproduce" yet)
t=5 … 11 + 5x2 =21 (the new 6 aren't ready to "reproduce" yet)
t=6 … your turn ...
 

Ryan$

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t=0 ... 1 mosquito
t=2 … original 1 mosquito + 2 mosquitoes = 3 mosquitoes
t=3 … 3 mosquitoes + 1x2 mosquitoes = 5 mosquitoes (the new two aren't ready to "reproduce" yet)
t=4 … 5 mosquitoes + 3x2 mosquitoes = 11 mosquitoes (the new two aren't read to "reproduce" yet)
t=5 … 11 + 5x2 =21
t=6 … 21 + 5x2 =31
t=7 ....31+ 5x2 =41
t=8 .... 41 + 5x2 =51

right? I'm not totally confident about my answer because I still miss the concept of how you're multiplying at each step and what to include or not include of mosquitoes at each step still confused.
 

Ryan$

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at t=3 , we have three mosquitoes, so we must do 32 because each one is generate two mosquitoes .. I didn't understand why you're not including them .. if at t=3 I've already 3 mosquitoes as why not doing multiply 32 because each of them will generate two mosquitoes ..
 

Harry_the_cat

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Drawing it out might help. Use M for a new mosquito, M for a one-second old mosquito and M for a mature mosquito (ie more than 2 seconds old)
t=0 … M
t=1 … M (the original one is now 1 second old)
t=2 … M M M (the original one has matured and produced 2 more babies)
t=3 …
M M M M M (the original one is red the two from the time before are now 1 sec old and the original one produce 2 more babies)
t=4 …
M M M M M M M M M M M (the first is the original, the 2nd and 3rd are the babies from t=2 and now mature, the 4th and 5th are the babies from t=3 (now 1 sec old), and the last 6 are the new offspring from the 3 mature mosquitoes)
etc...
 

Harry_the_cat

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at t=3 , we have three mosquitoes, so we must do 32 because each one is generate two mosquitoes .. I didn't understand why you're not including them .. if at t=3 I've already 3 mosquitoes as why not doing multiply 32 because each of them will generate two mosquitoes ..
after two seconds he starts to generate at every second "two mosquito"

The way I read this is that the mosquito has to be 2 second old ("mature") before it generates more.
Once it is "mature" , it generates 2 mosquitoes every second.
 

Ryan$

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So at t=8 ; I have 30 mosquito?
 

Harry_the_cat

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Did you continue the process?
 

Ryan$

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Yes ofcourse .. It's like fibonacii sequence..
 

Harry_the_cat

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I haven't completed the problem myself but I'm guessing the answer I going to be more than 30.
 

Ryan$

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So please how could I be verified of my answer? it would be just a honor please if you can help me!
 

Harry_the_cat

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So please how could I be verified of my answer? it would be just a honor please if you can help me!
t=4 … M M M M M M M M M M M
What's the next row? It's your job to do the work, not mine!
 

Ryan$

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I solved the problem , thanks alot! it's like fibonacii problem..

at t=5 => 21
at t=6 => 43 (21+11*2)
at t=7 => 85(43 +21*2)
 

Ryan$

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what's confused me right now, if they asked what's mosquitoes at t=8 , will I include what's generated at t=8? which means the result should be the number of mosquitoes after t=8?
 

Ryan$

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here's other question to prove that I understand, lets assume now that at every two seconds they will be generated one mosquitoes, how many mosquitoes will be at 6.5(at 6) ?
here's my answer and re-correct me if I'm wrong please! :
t=0 ... 1 mosquito
t=1 .... 1+1
t=2 ..... 2+1*1
t=3 ..... 3+2*1
t=4 .... 5+3*1
t=5 .... 8+5*1
t=6 .... 13+(8*1) = 21

so at t=6 there're 21 mosquitoes ... right?!
 

Dr.Peterson

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That is not the answer to the problem as you stated it. It would be correct if every second, each mosquito more than one second old produced one more.

Do you see the two differences?
 

Ryan$

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But it's not telling me that; it tells me that once created mosquito at t=0 then at every more one second the current mosquitoes will generate one more mosquito
 
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