Euchre uses only 9’s, 10’s, Jacks, Queens, KIngs and aces. What is the probability that a 5 card hand will have either both red jacks or both black jacks?
So there are only 24 cards
number of 5 card hands = C(24,5) = ...
number containing both red Jacks = C(22,3) = ...
number containing both black jacks = C(22,3) =
number of containing all 4 jacks = C(20,1) = ....
number containing either both 2 black or 2 red Jacks
= C(22,3) + C(22,3) - C(10,1) = ..... **
prob of your stated event = **/C(24,5) = ....
Oops. Go with Reiny. My calculations were based on drawing only two cards.
24 cards in all, so
P(both red) = P(both black) = 1/12 * 1/23
so, ...
To find the probability of getting either both red jacks or both black jacks in a 5-card hand in Euchre, we first need to determine the total number of favorable outcomes and the total number of possible outcomes.
Total number of favorable outcomes:
To have both red jacks in the hand, we must choose the two red jacks out of the four available red jacks (there are two red jacks for each suit). This can be done in (4 choose 2) = 6 ways.
Similarly, to have both black jacks in the hand, we need to choose the two black jacks out of the four available black jacks. This can also be done in 6 ways.
Total number of possible outcomes:
To determine the total number of possible outcomes in a 5-card hand from a deck where only 9’s, 10’s, Jacks, Queens, Kings, and Aces are used, we need to consider the total number of choices for each card:
- 9’s: There are 4 choices available (one for each suit).
- 10’s: There are 4 choices available (one for each suit).
- Jacks: There are 4 choices available (two red jacks and two black jacks).
- Queens: There are 4 choices available (one for each suit).
- Kings: There are 4 choices available (one for each suit).
- Aces: There are 4 choices available (one for each suit).
Therefore, the total number of possible outcomes is 4^5 = 1024.
Now, we can calculate the probability:
Probability = (Total number of favorable outcomes) / (Total number of possible outcomes)
P(both red jacks or both black jacks) = (Number of ways to have both red jacks + Number of ways to have both black jacks) / (Total number of possible outcomes)
P(both red jacks or both black jacks) = (6 + 6) / 1024
P(both red jacks or both black jacks) = 12 / 1024
P(both red jacks or both black jacks) ≈ 0.0117
So, the probability that a 5-card hand in Euchre will have either both red jacks or both black jacks is approximately 0.0117, or 1.17%.