Question

Cards are:
Black A, K, Q, J
Red    A, K, Q, J

A

K

Q

J

A

K

Q

J

1.) The probability of picking a numeric card if you pick two cards randomly

2.) The probability of Picking K and Q if you pick two cards randomly. It does not matter which color. However, K needs to be in your left and Q needs to be in your right hand.

3.) The probability of Picking K and Q if you pick two cards randomly. It does not matter which color and which card you have in which hand.

4.) The probability of picking non-numeric cards if you pick two cards randomly one by one without putting the first one back

5.) If you pick a card what is the probability that you are not going to pick a “J”

6.) From the card above, if you pick three cards one after another without putting them back. What are the chances you are going to pick Red A, K, and Q?

Answer 1

Solution

Concept Base

A standard deck of cards has 52 cards. ………………………………………………………… (1)

Of these, 36 are number (numeric) cards bearing numbers 2 to 10 both inclusive – 4 cards of each number. ………………………………………………………………………………………….. (2)

The remaining 16 cards, called face cards, are made up of 4 cards each of J, Q, K and A ………(3)

Number of ways of selecting r things out of n things is given by ⁿC_r = (n!)/{(r!)(n - r)!}…….…(4)

Values of ⁿC_r can be directly obtained using Excel Function: Math & Trig COMBIN(Number, Number_chosen) [Number is n, Number_chosen is r]…………………………………………. (4a)

Probability of an event E, denoted by P(E) = n/N ………………………………..……………(5)

where n = n(E) = Number of outcomes/cases/possibilities favourable to the event E and N = n(S) = Total number all possible outcomes/cases/possibilities.

Now to work out the solution,

1.) Probability of picking a numeric card if you pick two cards randomly = 0.4344 Answer 1

Out of the two cards picked, one must be numeric and the other must be non-numeric. Vide (2), (3) and (4), the possible combinations are: ³⁶C₁ x ¹⁶C₁ = (36 x 16).

So, vide (5), n = (36 x 16).

Vide (4) again, 2 out of 52 cards can be picked in ⁵²C₂ = 1326 and vide (5), this is N.

Thus, vide (5), required probability = (36 x 16)/1326 = 0.4344.

2.) Probability of Picking K and Q if you pick two cards randomly so that K is in your left and Q is in your right hand = 0.0121 Answer 2

There are four K cards and four Q cards. The left card can be picked in 4 ways and the right also can be picked in 4 ways. Thus, n = 4 x 4 = 16 and N = 1326 as above. Then, the required probability

= 0.0121

3.) In this case, the order does not make a difference and hence it can be KQ or QK. Clearly,the probability = 2 x 0.0121 = 0.0242 Answer 3

4.) Vide (3), the first non-numeric card can any one of 16 and then the second can non-numeric only in the remaining 15 waays. Thus, vide (5), n = 16 x 15 and N = 52 x 51. So, vide (5),

Probability of picking non-numeric cards if you pick two cards randomly one by one without putting the first one back = 240/2652 = 0.0905 Answer 4

5.) Out of 52 cards, there are four J’s and hence 48 are non-J’s. So, probability of not picking a “J”

= 48/52

= 0.9231 Answer 5

6.) Out of 8 cards, there is one Red A, one Red K, and one Red Q. So, vide (5), n = 1 x 1 x 1 = 1.

N = 8 x 7 x 6 = 336

So, probability = 1/336

= 0.0030 Answer 6

DONE