What is the approx. probability that at least two of the first five pe

Question

A Tv show host picks members of the large studio audience at random and asks them what star sign they were born under.
(There are 12 star signs in all and you may assume that the probabilities that a randomly chosen person will be born under each star sign are equal.)

What is the approx. probability that at least two of the first five people picked were born under the same star sign?

A 0.60
B 0.50
C 0.62
D 0.38
E 0.55

learnwithrohan · Answer

Total Probability - No one born under the same sign
1 - none born 
1- (12/12) *(11/12)*(10/12)*(9/12)*(8/12)
1- 0.38 
0.62

arronjam · Answer

Hi can you comment on the solution that i provided though that solution does not give the correct answer?

sidsst · Answer

If every person picked up has a different start sign :

then 5 people could be picked up with probability : 12*11*10*9*8/12*12*12*12*12 { after picking one start sign your pool of star signs reduces by 1).
 = 0.38

Now if all people do not have different star signs then at least 2 of them must have same star sign. 
Thus probability that at least 2 people have same star signs = 1 - (probability that all have different star signs)
 = 1-0.38 =  0.62

Option C

learnwithrohan · Answer

Hey in the first STEP SSDDD it is actually SSDEF , what I mean is all the other signs are different hence you can not take all of them as D therefore , when you multiply by the final permutation is should be 5!/2! similiarly in second step it is 5!/3! only Hope it helps

arronjam · Answer

I think when we multiply by 5! we are double counting the permutations so it must be 5/3!2! 3 for removing the double counting and 2 for the same star signs SS?

arronjam · Answer

Hi  Bunuel

Can you help me get what is wrong in the following solution?

Hi All

Can someone help me with the following question?

I can solve it with the 1-complement approach but i want to solve for individual cases and add them up.
Case 1. Two people born under the same star.

SSDDD S stands for the same star sign and D for a different star sign.

12/12* 1/12* 11/12* 10/12* 9/12 * 5!/3!*2! =9900/12^4
5 stars can be arranged in 5! ways. we need to dived by 3! because the arrangement of three stars has already been counted and divide by 2 to remove the two same star signs eg AA. Is the reasoning correct here?

Similarly Other cases 
SSSDD
12/12 *1/12* 1/12*11/12 * 10/12 * 5!/3!*2! = 1100/12^4
SSSSD 
12/12 *1/12* 1/12* 1/12 * 11/12 * 5!/4! = 55/12^4
SSSSS 
12/12 *1/12* 1/12* 1/12 * 1/12 = 1/12^4

Adding up all these 4 cases yields 11056/12^4 = 0.53 but that is not the correct answer. Please help me get the mistake. Thank you

learnwithrohan · Answer

@aaronjam hey Sorry, I misread what you had done. So far you are absolutely correct, but you are missing out on two conditions Case 1(when two people are born under the same sign and the remaining three people are also born under the same sign, but different from the first sign i.e Jan Jan Feb Feb Feb) Case 2 (when two people are born under the same sign, two out of the remaining three are born under the same sign, but different from the first sign, and the last is born under a different sign from these four i.e Jan Jan Feb Feb Mar) Once you add these two cases you will get the correct solution. Hope this helps

What is the approx. probability that at least two of the first five pe

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What is the approx. probability that at least two of the first five pe

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