public double?GetExpectedNumberOfTurnsToPaintBalls(int numBalls)
{
if (this._numBallsToProbabilityDictionary.ContainsKey(numBalls))
{
return(this._numBallsToProbabilityDictionary[numBalls]);
}
if (numBalls <= 0)
{
return(null);
}
if (numBalls == 1)
{
return(0);
}
var initialStateBallDistribution = new Dictionary <int, int> {
{ 1, numBalls }
};
var initialState = new RandomBallPainterState(initialStateBallDistribution, numBalls);
return(this._markovChainSolver.GetExpectedValueOfNumTurnsToReachTerminalState(initialState, GetStateTransitions));
}
/// <summary>
/// returns a tuple of (state, probability transition to this state)
/// </summary>
/// <returns></returns>
public Dictionary <RandomBallPainterState, double> GetStateTransitions(RandomBallPainterState state)
{
Dictionary <RandomBallPainterState, double> stateProbabilityTransition = new Dictionary <RandomBallPainterState, double>();
foreach (var key1 in state.BallDistributionForState.Keys)
{
var probabilityFirstTypeOfBallDrawn = ((double)key1 * (double)state.BallDistributionForState[key1]) / (double)state.NumBalls;
foreach (var key2 in state.BallDistributionForState.Keys)
{
if (key1 == key2)
{
// if we've drawn a second ball that has the same number of balls of the same color, there is a chance that
// we've drawn the same colored ball twice if we're looking at a color with > 1 ball in it
var probabilityOfDrawingSameBallTwice = (key1 - 1) / ((double)state.NumBalls - 1);
var currentStateClone = new RandomBallPainterState(state.DeepCloneBallDistribution(), state.NumBalls);
stateProbabilityTransition[currentStateClone]
= (stateProbabilityTransition.ContainsKey(currentStateClone) ? stateProbabilityTransition[currentStateClone] : 0)
+ probabilityFirstTypeOfBallDrawn * probabilityOfDrawingSameBallTwice;
if (state.BallDistributionForState[key2] == 1)
{
continue;
}
// it's also possible we drew a ball of a different color that happened to have the same number of balls of the same color
// as the first ball that was drawn
var probabilityOfDrawingDifferentBall
= (double)key2 * ((double)state.BallDistributionForState[key2] - 1) / ((double)state.NumBalls - 1);
var nextStateDistribution = state.DeepCloneBallDistribution();
// if this happened, the second ball's group will lose 1 ball, and the first ball's group will gain 1 ball
// so we increment the value for key1+1 by 1, increment the value for key1-1 by 1, decrement the value for key1 by 2
nextStateDistribution[key1 + 1] = (nextStateDistribution.ContainsKey(key1 + 1) ? nextStateDistribution[key1 + 1] : 0) + 1;
nextStateDistribution[key1] = nextStateDistribution[key1] - 2;
// if there are no longer values for this key, remove it
if (nextStateDistribution[key1] == 0)
{
nextStateDistribution.Remove(key1);
}
if (key1 - 1 > 0)
{
nextStateDistribution[key1 - 1] = (nextStateDistribution.ContainsKey(key1 - 1) ? nextStateDistribution[key1 - 1] : 0) + 1;
}
var nextState = new RandomBallPainterState(nextStateDistribution, state.NumBalls);
stateProbabilityTransition[nextState]
= (stateProbabilityTransition.ContainsKey(nextState) ? stateProbabilityTransition[nextState] : 0)
+ probabilityFirstTypeOfBallDrawn * probabilityOfDrawingDifferentBall;
}
// if we drew a second type of ball
else
{
var probabilityDrawingSecond
= ((double)key2 * (double)state.BallDistributionForState[key2]) / ((double)state.NumBalls - 1);
// in this case, the first ball's and second ball's groups will lose 1
// the first ball's group + 1 will gain 1 and the second ball's group - 1 will gain 1
var nextStateDistribution = state.DeepCloneBallDistribution();
nextStateDistribution[key1] = nextStateDistribution[key1] - 1;
if (nextStateDistribution[key1] == 0)
{
nextStateDistribution.Remove(key1);
}
nextStateDistribution[key2] = nextStateDistribution[key2] - 1;
if (nextStateDistribution[key2] == 0)
{
nextStateDistribution.Remove(key2);
}
nextStateDistribution[key1 + 1] = (nextStateDistribution.ContainsKey(key1 + 1) ? nextStateDistribution[key1 + 1] : 0) + 1;
if (key2 - 1 > 0)
{
nextStateDistribution[key2 - 1] = (nextStateDistribution.ContainsKey(key2 - 1) ? nextStateDistribution[key2 - 1] : 0) + 1;
}
var nextState = new RandomBallPainterState(nextStateDistribution, state.NumBalls);
stateProbabilityTransition[nextState]
= (stateProbabilityTransition.ContainsKey(nextState) ? stateProbabilityTransition[nextState] : 0)
+ probabilityFirstTypeOfBallDrawn * probabilityDrawingSecond;
}
}
}
return(stateProbabilityTransition);
}
