Both Alexa and Shelly have an infinite bitstring. Alexa’s bitstring is denoted by a1a2a3 . . ., whereas Shelly’s bitstring is denoted by s1s2s3 . . .. Alexa can see her bitstring, but she cannot see Shelly’s bitstring. Similarly, Shelly can see her bitstring, but she cannot see Alexa’s bitstring. The bits in both bitstrings are uniformly random and independent. The ladies play the following game: Alexa chooses a positive integer k and Shelly chooses a positive integer l. The game is a success if sk = 1 and al = 1. In words, the game is a success if Alexa chooses a position in Shelly’s bitstring that contains a 1, and Shelly chooses a position in Alexa’s bitstring that contains a 1
• Assume Alexa chooses k = 4 and Shelly chooses l= 7. Determine the probability that the game is a success.
• Assume Alexa chooses the position, say k, of the leftmost 1 in her bitstring, and Shelly chooses the position, say l of the leftmost 1 in her bitstring. –
If k is not equal to l, is the game a success? –
Determine the probability that the game is a success.