I was wondering if someone could help me check my work on this problem and see if I'm correct.
Consider the experiment of drawing two cards without replacement from an ordinary deck of 52 playing cards.
What are the odds in favor of drawing a spade and a heart?
There are 52 playing cards in a deck
There are 13 spades and 13 hearts in deck
13 + 13 = 26
P(spade and heart) = 26/52
P(not a spade or heart) = 26/52
Odds in favor of A = Number of ways that A could occur/Number of ways that A could not occur
Odds in favor of A = 26/52 / 1- 26/52 = 0
Or would it be
Odds in favor of A = 26/52 / 1 - 26/52 = 26 / 51
"Odds in favor of A = 26/52 / (1- 26/52)= 0"
A in the above expression actually evaluates to 26/52 / (1-26/52) = (1/2) / (1/2) =1:1
However, I would reason it this way:
The first card can be either a spade or a heart, so the probability is 26/52.
The second card has to the complementary suit, namely a spade if the first one was a heart, and vice versa.
Probability of the second card is therefore limited to one suit out of 51 cards, or 13/51.
Probability of both events happening is
(26/52)*(13/51)=(1/2)*(13/51) = 13/102
Odds are
(13/102) / (1-13/102)
= 13/89
13:89
I have a question about the answer, how did you get 13/89, I don't understand how you got 89
Here it is, in a little more detail:
(13/102) / (1-13/102)
= (13/102) / ((102-13)/102)
=(13/102) / (89/102)
=13/89
=13:89
To check if your work is correct, let's go through the problem step by step.
First, you correctly identified that there are 52 playing cards in a deck, with 13 spades and 13 hearts. This means that the total number of favorable outcomes (spade and heart) is 13 + 13 = 26.
Next, you calculated the probability of drawing a spade and a heart as 26/52. This is correct because the probability is calculated by dividing the number of favorable outcomes (26) by the total number of outcomes (52).
Then, you calculated the probability of not drawing a spade or a heart as 26/52. This is also correct, as there are 26 cards (clubs and diamonds) that are not spades or hearts. This probability is equivalent to 1 minus the probability of drawing a spade and a heart.
Finally, you tried to calculate the odds in favor of drawing a spade and a heart. However, there seems to be an error in your calculation. The correct formula for odds in favor of A is:
Odds in favor of A = Number of ways that A could occur / Number of ways that A could not occur
In this case, the number of ways that drawing a spade and a heart could occur is 26 (since we have already established that there are 26 favorable outcomes). The number of ways that drawing a spade and a heart could not occur is 52 minus 26, which is 26. Therefore, the correct calculation would be:
Odds in favor of drawing a spade and a heart = 26/26 = 1
So, the odds in favor of drawing a spade and a heart are 1. This means that there is only one possible outcome that satisfies the condition, out of a total of one possible outcome that does not satisfy the condition.
In conclusion, your work is correct except for the last calculation. The correct odds in favor of drawing a spade and a heart are 1, not 0 as you initially calculated.