Sometimes a reward isn't a fixed number.
It can change depending on an outcome, like drawing a card.
Let's look at this first choice. A guaranteed one point for a circle card or the face value of a drawn card with a circle. Drawing any card from this deck is guaranteed to get a circle or one point.
However, if we draw a card with a circle and a two, then we get two points. Since there are two cards greater than one, drawing a face card would be a more valuable goal on average. Here's another comparison. We can get a fixed two points for drawing a square. Or the face value of a triangle card.
Since there are two triangles and two squares, the probability of drawing a square or a triangle is the same.
However, a triangle card would reward more points on average. So, it's the more valuable goal.
Now, let's compare two different variable rewards. For the first goal, we only get a reward if the card's value is two or less. Out of the four possible cards to draw, our rewards would be one point, two points, 0 points, and zero points. Again, the average is 1 + 2 / 4, which is 3/4 of a point. The second goal gives the full face value. if you draw a star. So the possible rewards are zero points, two points, zero points, and four points. The average is 2 + 4 / 4, which is 1 point and a half. So the second goal is more valuable.
With variable rewards, the key is to consider each case where the goal would be met.
Let's try another. Here we choose between a guaranteed four points for drawing a four or the face value of a circle card. There are two fours in the deck and two circle cards worth two points and four points respectively.
Since the probability of drawing a circle card or a four is equal, the fours are a more valuable reward because they award more points on average.
This next comparison is between two similar goals. Drawing a square versus drawing a star. Drawing a square awards one point, zero points, zero points, or four points. Drawing a star awards zero points, one point, two points, or zero points. Since the probability of drawing a square and a star is the same, the square is a more valuable reward because it awards more points on average. The average amount of points awarded for drawing a square is 5 / 2 or 2.5 points.
The average amount of points awarded for drawing a star is 3 / 2 or 1.5 points.
The square awards more points on average, so it is the more valuable goal.
