232 I deal myself 3 cards from a standard 52card deck Find t

2.32 I deal myself 3 cards from a standard 52-card deck. Find the probabilities of each of the following events: (a) 2 of a kind (e.g., 2 fives or 2 kings); (b) 3 of a kind; (c) 3 of the same suit (a.k.a a flush, e.g., 3 hearts or 3 clubs); (d) 3 cards in consecutive order (a.k.a. a straight, e.g., 2-3-4 or 10-J-Q).

Solution

a)
Selecting any one kind = 13C1
So, for two of that kind , second card can have 4 possibilities out of that kind , i.e = 4C2
Remaining one card can be out of these cases = 48C1
P(2 of a kind)= (13C1*(4C2)*48C1)/52C3 = 0.1694 Answer

b)
P(3 of a kind)= (13C1*4C3)/52C3 = 0.0023 Answer

c)
Selecting any one suit = 4C1
So, for three of that suit , second card can have 13 possibilities out of that kind , i.e = 13C3
P(3 of same suit) = (4C1 * 13C3)/52C3 = 0.0517 Answer

d)

Only possible numbers are : A,2,3,....,J.
Therefore,
Selecting one number out of these =11C1 *4C1
Now,second and third can be the change in suits of the consecutice numbers i.e = 4C1*4C1
P(3 cards in consecutive order)=(11C1 * 4C1 * 4C1 *4C1) / 52C3 =0.0318 Answer