Probability Question?

Question

A fair die is tossed repeatedly and each outcome is recorded. What is the expected number of tosses before the first repeat occurs?

Answer 1

Hint: count the number with no repeats.

Roll 1: 6 outcomes with no repeats, 0 with repeats
Roll 2: 5 outcomes with no repeats, 1 with repeat
Roll 3: 4 outcomes with no repeats, 2 with repeat
Roll 4: 3 outcomes with no repeats, 3 with repeat
Roll 5: 2 outcomes with no repeats, 4 with repeat
Roll 6: 1 outcomes with no repeats, 5 with repeat
Roll 7: 0 outcomes with no repeats, 6 with repeat

1 roll: 6 out of 6
2 rolls: 6*5 out of 6^2 = 5/6
3 rolls: 6*5*4 out of 6^3 = 5/9
4 rolls: 6*5*4*3 out of 6^4 = 5/18
5 rolls: 6*5*4*3*2 out of 6^5 = 5/54
6 rolls: 6*5*4*3*2*1 out of 6^6 = 5/324
7 rolls: 6*5*4*3*2*1*0 out of 6^7 = 0

Answer 2

P(repeat on 2nd toss) =
1 - 5/6 = 0.167
P(repeat on 3rd toss) =
1 - (5/6)(4/6) = 0.444
P(repeat on 4th toss) =
1 - (5/6)(4/6)(3/6) = 0.722
P(repeat on 5th toss) =
1 - (5/6)(4/6)(3/6)(2/6) = 0.907
P(repeat on 6th toss) =
1 - (5/6)(4/6)(3/6)(2/6)(1/6) = 0.985
P(repeat on 7th toss) =
1 - (5/6)(4/6)(3/6)(2/6)(1/6)(0) = 1

Repeats may be reasonably expected from the 4th toss on. There will certainly be a repeat on the 7th toss, if not before.

Answer 3

after 6 tosses

Answer 4

after six tosses