What are the odds in favor of drawing a spade and a heart if drawing from a deck of cards w/o replacement?
the prob of a spade and a heart
= 2(13/52)(13/51) = 13/102
so the prob of NOT a spade and a heart
= 1 - 13/102
= 89/102
so the odds in favour of drawing a spade and a heart is 13:89
To determine the odds in favor of drawing a spade and a heart from a deck of cards without replacement, we need to consider the total number of favorable outcomes and the total number of possible outcomes.
Here's how you can calculate it:
1. Start by determining the total number of cards in the deck. In a standard deck of 52 playing cards, there are 13 cards of each suit (spades, hearts, diamonds, and clubs).
2. We need to calculate the number of favorable outcomes. In this case, we are looking for one spade and one heart.
- First, calculate the number of ways to choose one spade from 13 spades. This is denoted as C(13, 1) = 13.
- Next, calculate the number of ways to choose one heart from the remaining 13 hearts after one spade has been drawn. This is also C(13, 1) = 13.
Therefore, the number of favorable outcomes is 13 * 13 = 169.
3. Now, we need to calculate the total number of possible outcomes. In the beginning, there are 52 cards in the deck. After drawing one card, there are 51 cards left. So, the total number of possible outcomes is C(52, 1) * C(51, 1) = 52 * 51 = 2652.
4. Finally, we can calculate the odds in favor by dividing the number of favorable outcomes by the number of possible outcomes and then simplify the fraction, if necessary.
Odds in favor = Number of favorable outcomes / Number of possible outcomes
= 169 / 2652
To simplify the fraction, you can divide both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 169 and 2652 is 1. So, the simplified fraction is:
Odds in favor = 169 / 2652
Therefore, the odds in favor of drawing a spade and a heart from a deck of cards without replacement are 169 to 2652.