Question

A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is
i. A card of spade or an ace.
ii. A black king.
iii. Neither a jack nor a king
iv. Either a king or a queen.

Answer 1

Hint:First take the sample space as 52 and then use the formula of finding the probability i.e find the favorable outcome from the sample space. By taking the ratio of favourable outcomes and total outcomes we get the probability.Use the formula probability $=\dfrac{n\left( A \right)}{n\left( S \right)}$ to find the probability in each condition.

Complete step-by-step answer:
As we all know that a deck of playing cards contains 52 cards, and here the set of these cards is our sample space, therefore the number of sample spaces will become 52.
Therefore,
Total number of cards = n(S) = 52 …………………………………………. (1)
As we have our sample space therefore we can solve the problems one by one,

i) Here we have to find the probability that a card drawn is a card of spade or an ace.
Let, ‘A’ be the set of conditions given above.
As we all know there are four types of cards in a deck namely Hearts, Clubs, Diamonds and Spades and they all contain equal numbers of cards in the deck.
Therefore total number of spades $=\dfrac{52}{4}=13$
Also, each set has one Ace. As there are four sets therefore each one will contain one Ace i. e. we have a total four Aces.
As we have considered all spades therefore ultimately we have considered the Ace of spade also and therefore remaining Aces are three now.
Therefore,
Number of spades including all Aces = 13 + 3 = 16
$\therefore $ n(A) = 16 ……………………………………………….……………………… (2)
Now to proceed further in the solution we should know the formula of finding probability given below,
Formula:
Probability $=\dfrac{n\left( A \right)}{n\left( S \right)}$
If we put the values of equation (1) and (2) in the above formula we will get,
$\therefore $ Probability $=\dfrac{16}{52}$
$\therefore $ Probability$=\dfrac{4}{13}$
Therefore, the probability that a card drawn is a card of spade or an ace is$\dfrac{4}{13}$.

ii) Here we have to find the probability that a card drawn is a black king.
Let, ‘B’ be the set of the conditions given above.
As we all know that each type of card contains one king and there are two types of black cards namely Spades and Clubs.
Therefore total number of black kings = 2
$\therefore $ n(B) = 2 …………………………………………………………. (3)
Now to proceed further in the solution we should know the formula of finding probability given below,
Formula:
Probability $=\dfrac{n\left( B \right)}{n\left( S \right)}$
If we put the values of equation (1) and (3) in the above formula we will get,
$\therefore $ Probability $=\dfrac{2}{52}$
$\therefore $ Probability $=\dfrac{1}{26}$
Therefore the probability that a card drawn is a black king is$\dfrac{1}{26}$.

iii) Here we have to find the probability that a card drawn is neither a jack nor a king.
Let, ‘C’ be the set of conditions given above.
As we all know that the deck of cards has four sets of each card. Therefore there are four jacks and 4 kings are there in a deck.
As we want a condition where both jack and king are not required therefore we have to subtract them from the total number of cards.
Therefore Number of cards with neither jack or king = 52- (4+4)
Therefore Number of cards with neither jack or king = 44
$\therefore $ n(C) = 44 ………………………………………………… (4)
 Now to proceed further in the solution we should know the formula of finding probability given below,
Formula:
Probability $=\dfrac{n\left( C \right)}{n\left( S \right)}$
If we put the values of equation (1) and (4) in the above formula we will get,
$\therefore $ Probability $=\dfrac{44}{52}$
$\therefore $ Probability $=\dfrac{11}{13}$
Therefore the probability that a card drawn is neither a jack nor a king is$\dfrac{11}{13}$.

iv) Here we have to find the probability that a card drawn is either a king or a queen.
Let, ‘D’ be the set of conditions given above.
As we know
As we all know that the deck of cards has four sets of each card. Therefore there are four kings and four queens are there in a deck.
Therefore total number of kings and queens = 4 + 4 = 8
$\therefore $ n(D) = 8 ……………………………………………………. (5)
Now to proceed further in the solution we should know the formula of finding probability given below,
Formula:
Probability $=\dfrac{n\left( D \right)}{n\left( S \right)}$
If we put the values of equation (1) and (5) in the above formula we will get,
$\therefore $ Probability $=\dfrac{8}{52}$
$\therefore $Probability $=\dfrac{2}{13}$
Therefore the probability that a card drawn is either king or queen is$\dfrac{2}{13}$.

Note: There are chances that you might get confused in sub-problem, (iii) and (iv) but remember that in (iii) you don’t have ignore number of jack and king and in (iv) you have to consider only the number of cards of kings and queens in the deck.Student should know this concept while solving these types of problems.A standard deck of playing cards contains 52 cards.Divided equally into two colors "Red" and "Black".Deck of 52 cards has four suits "Spades", "Hearts", "Diamonds" and "Clubs".Hearts and Diamonds comes in Red color and Spades and Clubs comes in Black color.Each 4 suits contains 13 cards : Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.A king, queen, or jack of a deck of playing cards are known as face cards.