Odds: I can choose either a deck of 2 or 4 cards. The cards say what chance I have of succeeding. The deck of 2 cards has a 50% and 75%. The deck of 4 has 45% 50% 75% and 80%. Which deck should I pick to increase my odds for success?

Question

Odds:  I can choose either a deck of 2 or 4 cards.  The cards say what chance I have of succeeding.  The deck of 2 cards has a 50% and 75%.  The deck of 4 has 45% 50% 75% and 80%.  Which deck should I pick to increase my odds for success?

doc.brown · Answer

In the end I will have to choose only one card from the chosen deck.

kmeis002 · Answer

You need to calculate the Expected value of each deck, or what value you can expect if choosing from each deck.

$$EV= \sum x_ip_i   $$
where x= value of a card aand
p = probability of obtaining that card.

So for case 1 (2 card deck):
Our values for x are .5 and .75, and the probability of drawing each card is 0.5 so
$$EV_{deck1} = (0.5)(0.5) + (0.75)(0.5) = .625 $$

We can expect a value of 0.625, or in this case a 0.625% success rate

Deck 2, the values can be found by looking, but the probability decreases to 0.25:

$$ EV = (0.25)(0.45) + (0.25)(.5) + (0.25)(0.75) + (0.25)(0.8) = 0.625$$

So what do these results say about which deck we should choose?