What are the odds in favor of drawing a spade and a heart without replacement from an ordinary deck of 52 playing cards?

I assume that you are talking about first a spade and then a heart. 13/52 for drawing in the first draw and 13/51 for the second. The probability of all events occurring is found by multiplying the probability of the individual events.

The same is true for drawing the heart first. If you are considering either one possibility or the other, add the two probabilities found.

To determine the odds in favor of drawing a spade and a heart without replacement from an ordinary deck of 52 playing cards, we need to calculate the probability of this event occurring and then express it as odds.

First, let's consider how many spades and hearts there are in a standard deck. There are 13 spades and 13 hearts, making it a total of 26 cards of interest.

When we draw the first card, we have a 26/52 chance of getting either a spade or a heart since there are 26 cards of interest out of the total 52 cards.

After removing one card, there are 51 cards remaining in the deck. Out of those 51 cards, there will be 12 spades and 13 hearts left, since we already drew one card of interest.

So, when we draw the second card, we have a 25/51 chance of getting either a spade or a heart.

To calculate the overall probability, we need to combine the probabilities of drawing a spade first and a heart second with drawing a heart first and a spade second. Since these two scenarios are mutually exclusive, we can add their probabilities:

P(Spade first and Heart second) = (26/52) * (13/51)
P(Heart first and Spade second) = (13/52) * (12/51)

Finally, we can sum these probabilities:

P(Spade first and Heart second) + P(Heart first and Spade second) = (26/52) * (13/51) + (13/52) * (12/51) = 338/2652 ≈ 0.1275

To express this probability as odds, we can divide the probability of success by the probability of failure:

Odds in favor = (P(Success)) / (P(Failure)) = (P(Spade first and Heart second) + P(Heart first and Spade second)) / (1 - (P(Spade first and Heart second) + P(Heart first and Spade second)))

Substituting the values:

Odds in favor = (338/2652) / (1 - (338/2652)) ≈ 0.1275 / 0.8725 ≈ 0.1462

Therefore, the odds in favor of drawing a spade and a heart without replacement from a standard deck of 52 playing cards is approximately 0.1462.