Bill plays a game for three turns. On each turn, he either rolls a fair, six sided die or flips a fair coin. If he rolls a 1 or 2 on the die, he will switch to the coin on the next turn, and if he flips a tails on the coin, he will switch to the die on the next turn. If Bill starts by rolling the die, what is the probability that he will flip the coin on the third turn?

mathtoo  Mar 4, 2018

2+0 Answers


Dang.....I hate typing in my answer when I cannot see the question on my screen...I got the turn 3 backwards......so here it goes a gain.....

He has two option   roll flip flip    or  roll roll flip


1st roll has a 2/6 chance of being a 1 or a 2    2/6 = 1/3

  then he flips a coin and has a 1/2 chance it is a HEADS   (so he would flip coin again turn 3)


He has a 4/6 chance of NOT rolling a 1 or a 2 ...... which means turn 2 would be a roll again

  then he has a 2 out of 6  (1/3 chance of rolling a 1 or a two)  which makes turn 3 a coin flip


1/3 * 1/2    +   4/6 * 1/3 =  1/6 + 4/18 = 7/18 chance of coin flip on turn 3







This is my first answer that was a  WRONG answer:


Well.....no one else has posted an answer to this....so I will make my usual probability-guess at it!


He has 2 out of 6 chance of rolling a 1 or a 2   on the die......  2/6 = 1/3

    then when he flips the coin, he has a 1/2 chance of tails   (which would make turn 3 a roll of the die)


He has a 4/6 chance of NOT rolling a 1 or a 2  (which means he ROLLS again)

   His second turn will  be a roll.... and he has a 4/6 chance this will NOT be a 1 or a 2...

        which would make turn 3 a roll of the die


1/3 x 1/2    +   4/6 * 4/6  = 1/6 + 16/36 = 22/36 = 11/18 chance turn 3 will be a roll (??)

ElectricPavlov  Mar 4, 2018
edited by ElectricPavlov  Mar 5, 2018

As EP said above, it's hard to do these types of problems without an image to see. So, let's make an image.


He starts out with the dice, so he has a chance of 1-6.



When he rolls a 1 or a 2, he goes to the coin. Otherwise, he stays on the dice.



Now we can write which corresponds to what Bill's next turn will result in coin or dice



Now that there's an image, we can count how many chances go to the coin on that last turn.


There are 14 total outcomes by the 3rd move. 5 of them lead to using the coin on the last turn.


The probability of Billy using a coin on the last turn should be 5/14.


I don't know if this is right, since EP had 2 totally different answers than I got, so... xD

CoopTheDupe  Mar 6, 2018

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