Separate the 13 spades from a standard deck of cards. Three of them are picture

Question

Separate the 13 spades from a standard deck of cards. Three of them are picture cards: the Jack, Queen, and King. Now draw two cards at random from these 13 spades. Find the probability (rounded to the nearest .01%) that (a) both of the cards drawn are picture cards, if they are drawn i. without replacement. ii. with replacement. (b) at least one of the two cards drawn is a picture card, if they drawn i. without replacement. ii. with replacement. An 11-digit number is randomly chosen by drawing 11 times from a box that has one ticket for each of the numbers 0 to 9 and writing down numbers on the tickets in the order in which they are drawn. the Find the chance that exactly 2 of the digits in the number chosen are sevens.

Explanation / Answer

P(event) = No. Of Favourable outcomes/Total no. of outcomes

4) (a) 2 cards are drawn without replacement

Favourable outcomes : Drawing 1 card out of 3 from 13 = 3/13, and then drawing 1 card out of 2 remaining from the remaing 12 = 2/12

Therefore the required probability = 3/13 * 2/12 = 3/78 = 0.0385 = 3.85%

(b) 2 cards are drawn with replacement

Now in the first pick we probability of picking any 1 of 3 out of the 13 is 3/13. Then we replace

In the second pick, the probability is the same = 1/13

Since there is no specification of order therefore the number of ways of doing this 2! = 2

Therefore the required probability = (3/13)*(3/13) *2 = 18/169 = 0.1065 = 10.65%

5) (a) At least 1 card is a picture card without replacement

This is a combination of 2 different events:

(i) P(picking 1 picture card out of 3, and 1 non picture card out of 9)

1 picture card out of 3 = can be done in 3 ways and 1 normal card out of 13 can be done in 9 ways

Total ways = 13C2 = 78

Therefore required probability = 27/78

(ii) P( Picking both picture cards) which is 3/78 from 4(a)

Therefore the required probability = 27/78 + 3/78 = 30/78 = 0.3846 = 38.46%

(b) Picking at least 2 face cards with replacement

Again we break it into 2 parts

(i) Picking one picture card and one normal card

Favourable Outcomes: Picking a picture card = 3/13 and picking a non picture card = 9/13 (with replacement)

We must take into account the different arrangements = 2! (either picture card first or non picture card first)

Therefore the required probability for this event = 2 * (3/13) * (9/13) = 54/169

(ii) Picking up 2 picture cards, with replacement. This from 4(b) = 18/169

Therefore the required probability = 54/169 + 18/169 = 72/169 = 0.4260 = 42.6%

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