Marble probability

Question

There are the 10 marbles in a jar labelled 1-5 and A-E.

 

a.  If you pick one, replace it, and then pick another one, what are the odds you will end up with one letter and one number?  What are the odds the first pick will be a letter and the second a number?

 

b.  Same question, but what if you do not replace them?

 

c.  If you pick 3, one at a time, and replace each, what are the odds you will pick 2 letters and one number?  What are the odds the first two will be a letter and the third a number?

 

d.  Same question, but what if you do not replace them?

 

 

Thanks!!!!

Answer 1

Let L = Letter, N = Number 

 

On the 1st pick P(L) = 1/2 and P(N) = 1/2 

 

If you replace then P remains the same on the 2nd pick so 

 

P(LL) = (1/2)*(1/2) = 1/4

P(LN) = (1/2)*(1/2) = 1/4

P(NL) = (1/2)*(1/2) = 1/4

P(NN) = (1/2)*(1/2) = 1/4  

 

Of these 4 possible outcomes, 2 have 1 L and 1 N, they are P(LN) + P(NL) = 1/4 + 1/4 = 1/2 

The probability of picking L first and N second is P(LN) = 1/4 

 

If you do not replace then P changes on the 2nd pick 

 

P(LL) = (1/2)*(4/9) = 4/18

P(LN) = (1/2)*(5/9) = 5/18

P(NL) = (1/2)*(5/9) = 5/18  

P(NN) = (1/2)*(4/9) = 4/18 

 

P(LN) + P(NL) = 5/18 + 5/18 = 10/18 = 5/9 

P(LN) = 5/18 

 

If you pick 3 with replacement then 

 

P(LLL) = (1/2)*(1/2)*(1/2) = 1/8

P(LLN) = (1/2)*(1/2)*(1/2) = 1/8

P(LNL) = (1/2)*(1/2)*(1/2) = 1/8

P(LNN) = (1/2)*(1/2)*(1/2) = 1/8

P(NLL) = (1/2)*(1/2)*(1/2) = 1/8

P(NLN) = (1/2)*(1/2)*(1/2) = 1/8

P(NNL) = (1/2)*(1/2)*(1/2) = 1/8

P(NNN) = (1/2)*(1/2)*(1/2) = 1/8

 

3 of these have 2 letters and 1 number, they are P(LLN) + P(LNL) + P(NLL) = 3/8 

P(LLN) = 1/8 

 

If you pick 3 without replacement then 

 

P(LLL) = (1/2)*(4/9)*(3/8) = 12/144

P(LLN) = (1/2)*(4/9)*(5/8) = 20/144

P(LNL) = (1/2)*(5/9)*(4/8) = 20/144

P(LNN) = (1/2)*(5/9)*(4/8) = 20/144

P(NLL) = (1/2)*(5/9)*(4/8) = 20/144

P(NLN) = (1/2)*(5/9)*(4/8) = 20/144

P(NNL) = (1/2)*(4/9)*(5/8) = 20/144

P(NNN) = (1/2)*(4/9)*(3/8) = 12/144 

 

P(LLN) + P(LNL) + P(NLL) = 20/144 + 20/144 + 20/144 = 60/144 = 5/12 

P(LLN) = 20/144 = 5/36