Question

A card is selected from a pack of 52 cards
A. How many points are there in the sample space?
B. Calculate the probability that the card is an ace of spades.
C. Calculate the probability that the card is (i) an ace (ii) black card.

Answer 1

Hint: In this type of questions, we should know the different types of cards in the pack of cards, i.e., 52, in a pack of 52 cards there will be four different types of cards, they are hearts, diamonds, club and spades, and each of these types have 13 cards in each and in 52 pack of cards there will be 26 are in colour black and 26 are in red colour, with this information and by using the probability formula
Probability=\[\dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{number}}\,\,{\text{of}}\,\,{\text{total}}\,\,{\text{outcomes}}}}\].

Complete step-by-step answer:
Given a card is selected from a pack of 52 cards, and in a pack of 52 cards there will be four different types of cards, they are hearts, diamonds, club and spades, and each of these types have 13 cards in each and in 52 pack of cards there will be 26 are in colour black and 26 are in red colour, in which diamonds and hearts are in red colour and spades and club are in black colour, there will numbering from 1 to 10 and there will be also face cards such as king , queen and jack, and an ace card which will also be considered as number 1.
Now we have to find the probability given, using the probability formula which is given by,
Probability=\[\dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{number}}\,\,{\text{of}}\,\,{\text{total}}\,\,{\text{outcomes}}}}\],
Here we have to find the points in the sample space, when a card is selected from a pack of 52 cards, so the number of possible outcomes are 52, and the sample space contains 52 elements, so finally 52 points will be there in the sample space.
Here we have to find the probability that the card is an ace of spades, we know that there will be only one ace spade in a pack of 52 cards,
Now using the formula i.e.,
Probability=\[\dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{number}}\,\,{\text{of}}\,\,{\text{total}}\,\,{\text{outcomes}}}}\],
Here number of favourable outcomes = 1, and number of total outcomes = 52,
\[\therefore \]probability that the card is an ace of spade if a card is drawn from a pack of 52 cards = \[\dfrac{1}{{52}}\],
(i) Here we have to find the probability that the card is an ace, so, we know that there will be 4 aces in a pack of 52 cards, which are of spade, club, heart and diamond, so now the favourable outcomes will 4 and number of total outcomes will 52, so by using the formula we get,
Probability=\[\dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{number}}\,\,{\text{of}}\,\,{\text{total}}\,\,{\text{outcomes}}}}\],
Probability=\[\dfrac{{\text{4}}}{{52}}\]
\[ \Rightarrow \] Probability\[ = \dfrac{1}{{13}}\],
(ii) ) Here we have to find the probability that the card is a black, so, we know that there will be 26 black cards in a pack of 52 cards, , so now the favourable outcomes will 26 and number of total outcomes will 52, so by using the formula we get,
Probability=\[\dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{number}}\,\,{\text{of}}\,\,{\text{total}}\,\,{\text{outcomes}}}}\],
Probability=\[\dfrac{{26}}{{52}}\]
\[ \Rightarrow \] Probability\[ = \dfrac{1}{2}\].

Final Answer:
There will be 52 points in the sample space.
The probability that the card is an ace of spades is\[\dfrac{1}{{52}}\]
The probability that the card is an ace is \[\dfrac{1}{{13}}\] and the probability that the card is a black card is \[\dfrac{1}{2}\].

Note:
The probability of an event A is always greater or equal to zero but never be less than zero, if S is the sample space then the probability of occurrence sample space is always 1, that is if the experiment is performed then it is sure to get one of the sample spaces.