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Question: Bob repeatedly rolls a fair 6-sided die. What is the probability that he rolls his first 5 before he rolls his second (not necessarily distinct) even number?

 

What I've Tried: Let 'n' be the number of rolls in the game. This means that n has to be at least 2. I then created cases:

n = 2: Both rolls would have to be even so \(\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}\)

n = 3: We have to have an odd number (not a 5) and two even numbers (but one of the even numbers has to be last), so \(\frac{1}{2}\cdot\frac{1}{3}\cdot2\cdot\frac{1}{2} = \frac{1}{6}\)

 

After this, I'm not sure what to do. Help would be appreciated!

 Jun 1, 2021
 #1
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first 5 meaning he rolls the number 5, or rolls 5 times?

edited by MathProblemSolver101  Jun 1, 2021
 #2
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o = rolling a 1 or 3 (1/3 probability)

e = rolling a 2, 4, or 6 (1/2 probability)

f = rolling a 5 (1/6 probability)

 

So we have 3 cases. Rolling a bunch of "o"s before rolling an f. Rolling a bunch of "o"s and 1 "e" before rolling an f. Rolling an f. 

 

Rolling a bunch of "o"s before rolling an f.

((1/3) * 1/6) + ((1/3)^2 * 1/6) + ((1/3)^3 * 1/6).... + ((1/3)^infinity * 1/6)

This is a geometric sequence. 

a = 1/18, r = 1/3, a/(1-r)

(1/18)/(2/3) = 1/12

 

Rolling a bunch of "o"s and 1 "e" before rolling an f.

2((1/3) * 1/6) + 3((1/3)^2 * 1/6) + 4((1/3)^3 * 1/6)... + (infinity - 1)((1/3)^infinity * 1/6)

I'm not actually sure how to calculate this, but hopefully someone else can help with that. #It probably is around 1/5 to 1/4. 

 

Rolling an f. 

1/6

 

1/6 + 1/12 + (2((1/3) * 1/6) + 3((1/3)^2 * 1/6) + 4((1/3)^3 * 1/6)... + (infinity - 1)((1/3)^infinity * 1/6))

1/4 + (2((1/3) * 1/6) + 3((1/3)^2 * 1/6) + 4((1/3)^3 * 1/6)... + (infinity - 1)((1/3)^infinity * 1/6))

Sorry I wasn't able to help more. :((

 

=^._.^=

 Jun 1, 2021
 #3
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1 - Probability of rolling a "5" ==1/6. The expected probability of getting a "5" ==n * 1/6, solve for n.

 

n ==30 rolls.

 

2 - In 30 rolls, the expected probability of rolling a "2", "4", "6" ==30 * 1/2==15

 

3 - Then the probability of rolling a "5" before a "2", or a "4", or a "6" ==5 / 15 ==1/3

 Jun 1, 2021

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