In: Statistics and Probability
Have a regular deck of cards with no jokers (13 cards per suit, 4 suits) giving 52 cards. Suppose we draw 5 card hand, so 5 cards without replacement.
What is the probability of getting a royal flush but where the cards ordered by rank have alternate color? That is, order the cards as 10,J,Q,K,A and then check to see they have alternate colour. Note in a proper royal flush, it is all the one suit, but we have changed that to alternate colour. So, for example “red 10, black J, red Q, black K, red A” is OK but “red J, black 10, red Q, black K, red A” is not OK because once reordered in rank the alternating colour no longer holds. Note the order in which they are drawn from the pack is not considered. 2.2 No repeats.
What is the probability that in the sequence of cards, as they are drawn, no rank occurs twice in a row? So ignoring the suit, the following are allowed: A, 10, 4, J, 10 or A, 10, A, 4, A, but the following are not allowed: A, A, 10, 4, A (A repeated in positions 1 and 2), A, 4, 10, 10, J (10 repeated in positions 3 and 4
1. Five cards are to be drawn from the deck without replacement.
Total number of possibilities = = 2598960
Royal flush contains Five cards: 10, J, Q, K, A
The color has to be alternating as per the given question.
There are 2 types of red cards (Hearts and Diamonds) and two types of black cards (Clubs and Spades)
For choosing the first card : You can choose either of the 4 10's.
For choosing the second card: Based on the first card's color, we are left with 2 J's to choose from.
For choosing the third card: Based on the second card's color, we are left with 2 Q's to choose from.
For choosing the second card: Based on the third card's color, we are left with 2 K's to choose from.
For choosing the second card: Based on the fourth card's color, we are left with 2 A's to choose from.
Therefore, the total number of favourable conditions are : 4*2*2*2*2=64
So, the probability is given by
2. Five cards are to be drawn from the deck without replacement.
Total number of ways the cards can be picked. i.e. the permutations. = 52*51*50*49*48
Probability of getting no rank occur twice in a row = 1- probability of getting rank twice in a row
First chose one of the 4 positions and one card out of 13. Card can be arranged in these two spots in 2 ways.
Two positions are fixed. The rest three can be any card. i.e 50 cards, 49 cards, 48 cards to choose from.
Therefore the probability is
= .961