To verify SatoshiDICE games, look at the verifygame-page.js

https://satoshidice.com/js/verifygame-page.js

The random number is generated by the following seed:

seed = sha256(hexStringToByte(serverSeed + txid + voutHex))

randomNumber = getRandomNumber(seed)

function getRandomNumber(sha256) {
    var seed = new Array();
    for (var s = 0; s < 8; s++) {
        var intText = sha256.substring(s * 8, (s + 1) * 8);
        seed.push(parseInt(intText, 16));
        mersenneTwister.seedArray(seed);
    }
    return mersenneTwister.short();
}

When SatoshiDICE publishes the secret server seed, you can use that to verify that your random numbers were random. The output number is a 4 byte number in hex.

The random number generator is a Marsenne Twister returning a short int:

https://satoshidice.com/js/MersenneTwister.js

The hash of the secret seed is published a few days before, and always put on the blockchain to make sure nothing is changed: https://cashexplorer.bitcoin.com/tx/61ac980529ee745599ba02dae44bdcacba323e7cfd20214e84895d030883e2cd

You can see the secret server seeds here: https://satoshidice.com/provablyfair