More primes with one extra die roll

Player one rolls a fair die $10{,}000$ times and player two rolls one $10{,}001$ times. What is the probability that player two rolls strictly more prime numbers than player one?

$\tfrac12$. Compare after $10{,}000$ rolls each. By symmetry, two is ahead with the same probability as one, say $\tfrac{1-t}{2}$ where $t$ is the tie probability. Two wins if ahead already, or if tied and the extra roll is prime (probability $\tfrac12$, since 2, 3, 5 are prime): $\tfrac{1-t}{2}+\tfrac t2=\tfrac12$.