Question

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

Answer 1

Hint: In each, take the sample space as a pack of 52 cards, and for each part of the question given, find the number of the favorable cards of each and consider it as a number of favorable outcomes. Then find the probability using the formula \[\dfrac{\left( \text{Number of favorable outcomes} \right)}{\left( \text{Total number of outcomes that can occur} \right)}\].

Complete step-by-step answer:
So, here we are asked to find the probability of getting
(i) A king of red color
(ii) A face card
(iii) The jack of hearts
(iv) A red face card
(v) A spade
(vi) The queen of the diamond
At first, we will define what probability is and understand the basic terms related to probability, to be used in the question. The probability of an event is a measure of the livelihood that the event would occur. If an experiment’s outcome is equally likely to occur, then the probability of an event E is the number of outcomes in E divided by the number of outcomes in the sample space. Here, sample space consists of all the events that can occur possibly. So, it can be written as,
\[P\left( E \right)=\dfrac{n\left( E \right)}{n\left( S \right)}\]
Here, P(E) is the probability of an event or events, n(E) is the number of favorable events and n(S) is the number of all the events that can occur possibly. Now, in a pack of 52 cards, there are 4 suits available such as Spades, Heart, Club, and Diamond. All of them have 13 cards each. Each suit has 1 King, 1 Queen, 1 Jack, 1 Ace, and 9 cards numbered 2 – 10.
(i) The same space consists of 52 cards in the pack. So, n(S) = 52.
The number of favorable cards is 2, one which is King of Heart and the other is King of Diamond.
So, n(E) = 2.
So, the probability is \[\dfrac{n\left( E \right)}{n\left( S \right)}=\dfrac{2}{52}=\dfrac{1}{26}\].
(ii) The same space consists of 52 cards in the pack. So, n(S) = 52.
The number of favorable cards is 12, three cards King, Queen, and Jack in all four types of cards each.
So, n(E) = 12.
So, the probability is \[\dfrac{n\left( E \right)}{n\left( S \right)}=\dfrac{12}{52}=\dfrac{3}{13}\].
(iii) The same space consists of 52 cards in the pack. So, n(S) = 52.
The number of favorable cards is 1 as only one jack is there in the Hearts type.
So, n(E) = 1
So, the probability is \[\dfrac{n\left( E \right)}{n\left( S \right)}=\dfrac{1}{52}\].
(iv) The same space consists of 52 cards in the pack. So, n(S) = 52.
The number of favorable cards is 26, as there are 13 cards of Hearts and 13 cards of Diamonds.
So, n(E) = 26
So, the probability is \[\dfrac{n\left( E \right)}{n\left( S \right)}=\dfrac{26}{52}=\dfrac{1}{2}\].
(v) The same space consists of 52 cards in the pack. So, n(S) = 52.
The number of favorable cards is 13, as there are 13 cards of the type spades.
So, n(E) = 13.
So, the probability is \[\dfrac{n\left( E \right)}{n\left( S \right)}=\dfrac{13}{52}=\dfrac{1}{4}\].
(vi) The same space consists of 52 cards in the pack. So, n(S) = 52.
The number of favorable cards is 1, as only one queen of diamond exists.
So, n(E) = 1.
So, the probability is \[\dfrac{n\left( E \right)}{n\left( S \right)}=\dfrac{1}{52}\].

Note: Students should be knowing about the type and quantity of each type and color of the card. They should also know that the King, Queen, and the Jack are also collectively known as Face Cards. They should also know which symbol represents which card as well.