Answer:
a) 0.000001539
b) 0.00001385
c) 0.0002401
d) 0.001441
e) 0.001966
Step-by-step explanation:
Since there are 52 deck of cards, and the hand of the poker is 5 set, then the total number of hands achievable would be
T = 52! / 5!(52 - 5)!
T = 52! / 5! 47 !
T = 2598560 possibilities.
a) There are 4 ways of getting a royal flush. So the probability of getting a royal flush is
4 / 2598560 =
0.000001539
b) There are 9 hands from the 5 card hands, and also, there are 4 possible suits. So then, the probability is
9 * 4 / 2598560 =
36 / 2598560 =
0.00001385
c) There are 13 possible ways to get a four of a kind, since there are 5 cards with the poker, the remaining would be taken from the 48 remaining cards, thus
13 * 48 / 2598560 =
624 / 2598560 =
0.0002401
d) 3 of a kind in conjunction with a pair is needed to form a full house. This 3 of a kind can be gotten from any 4 suits. Then again, the pair has two cards with the same face value. So,
4! / 2! (4 - 2)! =
4! / 2! 2! = 6
That means, there are 6 possible ways to get our needed suits. Then, the probability of getting a full house is
13 * 4 * 12 * 6 / 2598560 =
3744 / 2598560 =
0.001441
e) To get a flush, all the 5 cards in the hand needs to have the same suit. Now, there are 13 different types of cards with only 5 cards being in the hand, thus
13! / 5! (13 - 5)! =
13! / 5! 8! = 1287
Now, recall that the question specifically asked us not to include any straight. There are 10 straights that can be gotten, and thus, we subtract it.
1287 - 10 = 1277
Since there are 4 suits, the probability of getting a flush is
1277 * 4 / 2598560 =
5108 / 2598560 = 0.001966
, 
, and so on.