Question

One card is drawn from a well-shuffled deck of $52$ cards. Find the probability of getting:
(i) a king of red suit (ii) a face card (iii) a red face card (iv) a queen of black suit (v) a jack of hearts (vi) a spade

Answer 1

No. of cards in pack $=52$

Solution(i):

No. of kings of red suit $=2$

Therefore, ${}^2{C}_1$( Selecting $1$ out of $2$ items) times out of ${}^{52}{C}_1$( Selecting $1$ out of $52$ items) a king of red suit is picked.

Let $E$ be the event of getting a kingof red suit from pack

We know that, Probability $P\left(E\right)$ $=\frac{\text{(No.of favorable outcomes)}}{\text{(Total no.of possible outcomes)}}$$=\frac{{}^2{C}_1}{{}^{52}{C}_1}$$=\frac{2}{52}$$=\frac{1}{26}$

Solution(ii):

No. of face cards $=12$

Therefore, ${}^{12}{C}_1$( Selecting $1$ out of $12$ items) times out of ${}^{52}{C}_1$( Selecting $1$ out of $52$ items) a face card is picked.

Let $E$ be the event of getting a face card from the pack

We know that, Probability $P\left(E\right)$ $=\frac{\text{(No.of favorable outcomes)}}{\text{(Total no.of possible outcomes)}}$$=\frac{{}^{12}{C}_1}{{}^{52}{C}_1}$$=\frac{12}{52}$$=\frac{3}{13}$

Solution(iii):

No. of red face cards $=6$

Therefore, ${}^6{C}_1$( Selecting $1$ out of $6$ items) times out of ${}^{52}{C}_1$( Selecting $1$ out of $52$ items) a red face card is picked.

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

We know that, Probability $P\left(E\right)$ $=\frac{\text{(No.of favorable outcomes)}}{\text{(Total no.of possible outcomes)}}$$=\frac{{}^6{C}_1}{{}^{52}{C}_1}$$=\frac{6}{52}$$=\frac{3}{26}$

Solution(iv):

No. of queens of black suit $=2$

Therefore, ${}^2{C}_1$( Selecting $1$ out of $2$ items) times out of ${}^{52}{C}_1$( Selecting $1$ out of $52$ items) a queen of black suit is picked.

Let $E$ be the event of getting a queen of black suit from pack

We know that, Probability $P\left(E\right)$ $=\frac{\text{(No.of favorable outcomes)}}{\text{(Total no.of possible outcomes)}}$$=\frac{{}^2{C}_1}{{}^{52}{C}_1}$$=\frac{2}{52}$$=\frac{1}{26}$

Solution(v):

No. of jacks of heart $=1$

Therefore, ${}^1{C}_1$( Selecting $1$ out of $1$ items) times out of ${}^{52}{C}_1$( Selecting $1$ out of $52$ items) a jack of hearts is picked.

Let $E$ be the event of getting a jack of hearts from pack

We know that, Probability $P\left(E\right)$ $=\frac{\text{(No.of favorable outcomes)}}{\text{(Total no.of possible outcomes)}}$$=\frac{{}^1{C}_1}{{}^{52}{C}_1}$$=\frac{1}{52}$

Solution(vi):

No. of spades $=13$

Therefore, ${}^{13}{C}_1$( Selecting $1$ out of $13$ items) times out of ${}^{52}{C}_1$( Selecting $1$ out of $52$ items) a spade is picked.

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

We know that, Probability $P\left(E\right)$ $=\frac{\text{(No.of favorable outcomes)}}{\text{(Total no.of possible outcomes)}}$$=\frac{{}^{13}{C}_1}{{}^{52}{C}_1}$$=\frac{13}{52}$$=\frac{1}{4}$