# 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

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Hint: Probability is given by number of favourable outcomes upon total number of outcomes.

As we know there are $52$ cards in total in a deck of cards.

Hence, total number of outcomes $ = 52$

(i) Probability of the card drawn is a card of spade or an Ace:

Total number of cards of Spade $ = 13$

Total number of Aces $ = 4$

Number of Aces of Spade $ = 1$

Therefore,

Probability of the card drawn is a card of spade or an Ace:

$\dfrac{{13}}{{52}} + \dfrac{4}{{52}} - \dfrac{1}{{52}} = \dfrac{{16}}{{52}} = \dfrac{4}{{13}}$ (We have subtracted 1 because the ace of spade is common)

(ii) Probability of the card drawn is a black king:

Total number of cards of black kings$ = 2$

Probability of the card drawn is a black king:

$\dfrac{2}{{52}} = \dfrac{1}{{26}}$ (since the black kings are only 2 )

(iii) Probability of the card drawn is neither a jack nor a king

Total number of jacks $ = 4$

Total number of kings $ = 4$

Probability of the card drawn is neither a jack nor a king:

$1 - \dfrac{{4 + 4}}{{52}} = \dfrac{{11}}{{13}}$

Total probability is 1 as always. Here we do not want jack or a king therefore we have subtracted the probability of getting it with total probability.

(iv) Probability of the card drawn is either a king or a queen:

Total number of Queens $ = 4$

Total number of Kings $ = 4$

Probability of the card drawn is either a king or a queen:

$\dfrac{{4 + 4}}{{52}} = \dfrac{2}{{13}}$

Since there are 4 kings and 4 queens .

Note :- In these types of questions we just have to apply the basic concepts of probability keeping in mind that we are not considering the same card twice. Here we have used nothing other than the number of favourable outcomes upon total number of outcomes.

As we know there are $52$ cards in total in a deck of cards.

Hence, total number of outcomes $ = 52$

(i) Probability of the card drawn is a card of spade or an Ace:

Total number of cards of Spade $ = 13$

Total number of Aces $ = 4$

Number of Aces of Spade $ = 1$

Therefore,

Probability of the card drawn is a card of spade or an Ace:

$\dfrac{{13}}{{52}} + \dfrac{4}{{52}} - \dfrac{1}{{52}} = \dfrac{{16}}{{52}} = \dfrac{4}{{13}}$ (We have subtracted 1 because the ace of spade is common)

(ii) Probability of the card drawn is a black king:

Total number of cards of black kings$ = 2$

Probability of the card drawn is a black king:

$\dfrac{2}{{52}} = \dfrac{1}{{26}}$ (since the black kings are only 2 )

(iii) Probability of the card drawn is neither a jack nor a king

Total number of jacks $ = 4$

Total number of kings $ = 4$

Probability of the card drawn is neither a jack nor a king:

$1 - \dfrac{{4 + 4}}{{52}} = \dfrac{{11}}{{13}}$

Total probability is 1 as always. Here we do not want jack or a king therefore we have subtracted the probability of getting it with total probability.

(iv) Probability of the card drawn is either a king or a queen:

Total number of Queens $ = 4$

Total number of Kings $ = 4$

Probability of the card drawn is either a king or a queen:

$\dfrac{{4 + 4}}{{52}} = \dfrac{2}{{13}}$

Since there are 4 kings and 4 queens .

Note :- In these types of questions we just have to apply the basic concepts of probability keeping in mind that we are not considering the same card twice. Here we have used nothing other than the number of favourable outcomes upon total number of outcomes.

Last updated date: 16th Sep 2023

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