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Question

A card is drawn at random from a pack of 52 cards. Find the probability that the card is drawn is

(i) a black king (ii) either a black card or a king (iii) a jack, queen or a king (iv) neither an ace nor a king (v) spade or an ace (vi) neither a red card nor a queen (vii) other than an ace (viii) a ten (ix) a spade (x) a black card (xi) the seven of clubs (xii) jack (xiii) the ace of spades (xiv) a queen (xv) a heart (xvi) a red card (xvii) neither a king nor a queen

(i) a black king (ii) either a black card or a king (iii) a jack, queen or a king (iv) neither an ace nor a king (v) spade or an ace (vi) neither a red card nor a queen (vii) other than an ace (viii) a ten (ix) a spade (x) a black card (xi) the seven of clubs (xii) jack (xiii) the ace of spades (xiv) a queen (xv) a heart (xvi) a red card (xvii) neither a king nor a queen

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Solution

No. of cards in a pack =52

**Solution(i):**

**Solution(iii):****Solution(iv):**

**Solution(****xvi****):**

No. of black kings =2

**Solution(ii):**

Therefore, 2C1( Selecting 1 out of 2 items) times out of 52C1( Selecting 1 out of 52 items) a black king is picked.

Let E be the event of getting a black kingfrom pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=2C152C1=252=126

No. of black cards or kings =28..... (26Black(including 2 Black kings) + 2 Red Kings)

Therefore, 28C1( Selecting 1 out of 28 items) times out of 52C1( Selecting 1 out of 52 items) a either a black card or a king is picked.

Let E be the event of getting either a black card or a king from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=28C152C1=2852=713

No. of jack, queen or king =12 (4- Jack, 4- Queen, 4-King)

Therefore, 12C1( Selecting 1 out of 12 items) times out of 52C1( Selecting 1 out of 52 items) a jack, queen or a king is picked.

Let E be the event of getting ajack, queen or a king from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=12C152C1=1252=313

No. of spade or ace =16 ........(13-Spade(including 1-Ace) + 3-Aces)

**Solution(v):**

Therefore, 16C1( Selecting 1 out of 16 items) times out of 52C1( Selecting 1 out of 52 items) a spade or an ace is picked.

Let E be the event of getting a spade or an ace from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=16C152C1=1652=413

No. of neither ace nor king=52−8=44 ...... (As there are 4-Kings and 4-Aces)

**Solution(vi):**

**Solution(vii****):**

**Solution(i****x****):**

Therefore, 44C1( Selecting 1 out of 44 items) times out of 52C1( Selecting 1 out of 52 items) a neither an ace nor a king is picked.

Let E be the event of gettingneither an ace nor a king from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=44C152C1=4452=1113

No. of neither red nor queen $=52-28=24$ ...(26-red+ 2-Black Queen)

Therefore, 24C1( Selecting 1 out of 24 items) times out of 52C1( Selecting 1 out of 52 items) neither red nor a queen is picked.

Let E be the event of gettingneither red nor queen from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=24C152C1=2452=613

No. of non-ace cards =52−4= 48 ......(4 Aces)

**Solution(viii****):**

Therefore, 48C1( Selecting 1 out of 48 items) times out of 52C1( Selecting 1 out of 52 items) non-ace card is picked.

Let E be the event of getting a non-ace card from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=48C152C1=4852=1213

No. of cards with no. 10 =4

Therefore, 4C1( Selecting 1 out of 4 items) times out of 52C1( Selecting 1 out of 52 items) card with no. 10 is picked.

Let E be the event of getting a card with no. 10 from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=4C152C1=452=113

No. of spade cards =13

**Solution(****x****):**

Therefore, 13C1( Selecting 1 out of 13 items) times out of 52C1( Selecting 1 out of 52 items) a spade card is picked.

Let E be the event of getting a spade card from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=13C152C1=1352=14

No. of black cards =26

**Solution(****xi****):**

Therefore, 26C1( Selecting 1 out of 26 items) times out of 52C1( Selecting 1 out of 52 items) a black card is picked.

Let E be the event of getting a black card from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=26C152C1=2652=12

No. of seven of clubs =1

**Solution(****xii****):**

Therefore, 1C1( Selecting 1 out of 1 items) times out of 52C1( Selecting 1 out of 52 items) a seven of clubs is picked

Let E be the event of getting a seven of clubs from the pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=1C152C1=152=152

No. of jacks =4

**Solution(****xiii****):****Solution(****xv****):**

Therefore, 4C1( Selecting 1 out of 4 items) times out of 52C1( Selecting 1 out of 52 items) a jack is picked.

Let E be the event of getting a jack from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=4C152C1=452=113

No. of ace of spades =1

**Solution(****xiv****):**

Therefore, 1C1( Selecting 1 out of 1 items) times out of 52C1( Selecting 1 out of 52 items) an ace of spade is picked

Let E be the event of getting an ace of spade from the pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=1C152C1=152=152

No. of queens =4

Therefore, 4C1( Selecting 1 out of 4 items) times out of 52C1( Selecting 1 out of 52 items) a queen is picked.

Let E be the event of getting a queen from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=4C152C1=452=113

No. of heart cards =13

Therefore, 13C1( Selecting 1 out of 13 items) times out of 52C1( Selecting 1 out of 52 items) a heart card is picked.

Let E be the event of getting a heart card from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=13C152C1=1352=14

No. of red cards =26

**Solution(****xvii****):**

Therefore, 26C1( Selecting 1 out of 26 items) times out of 52C1( Selecting 1 out of 52 items) a red card is picked.

Let E be the event of getting a red card from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=26C152C1=2652=12

No. of neither king nor queen cards =52−8=44 ...... (As there are 4-Kings and 4-Queen)

Therefore, 44C1( Selecting 1 out of 44 items) times out of 52C1( Selecting 1 out of 52 items) a neither king nor queen card is picked.

Let E be the event of gettingneither king nor queen card from pack

We know that, Probability P(E) =(No.of favorable outcomes)(Total no.of possible outcomes)=44C152C1=4452=1113

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