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.
(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.
Solution and Explanation
(a) When a card is selected from a pack of 52 cards, the number of possible outcomes is 52 i.e., the sample space contains 52 elements.
Therefore, there are 52 points in the sample space.
(b) Let A be the event in which the card drawn is an ace of spades.
Accordingly, n(A) = 1
\(∴(P)(A)=\frac{\text{Number of outcomes favourable to A}}{\text{Total number of possible outcomes}}=\frac{n(A)}{n(S)}=\frac{1}{52}\)
(c) (i)Let E be the event in which the card drawn is an ace.
Since there are 4 aces in a pack of 52 cards, n(E) = 4
\(∴(P)(E)=\frac{\text{Number of outcomes favourable to E}}{\text{Total number of possible outcomes}}=\frac{n(E)}{n(S)}=\frac{4}{52}=\frac{1}{!3}\)
(ii)Let F be the event in which the card drawn is black.
Since there are 26 black cards in a pack of 52 cards, n(F) = 26
\(∴(P)(F)=\frac{\text{Number of outcomes favourable to F}}{\text{Total number of possible outcomes}}=\frac{n(F)}{n(S)}=\frac{26}{52}=\frac{1}{2}\)
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Concepts Used:
Probability
Probability is defined as the extent to which an event is likely to happen. It is measured by the ratio of the favorable outcome to the total number of possible outcomes.
The definitions of some important terms related to probability are given below:
Sample space
The set of possible results or outcomes in a trial is referred to as the sample space. For instance, when we flip a coin, the possible outcomes are heads or tails. On the other hand, when we roll a single die, the possible outcomes are 1, 2, 3, 4, 5, 6.
Sample point
In a sample space, a sample point is one of the possible results. For instance, when using a deck of cards, as an outcome, a sample point would be the ace of spades or the queen of hearts.
Experiment
When the results of a series of actions are always uncertain, this is referred to as a trial or an experiment. For Instance, choosing a card from a deck, tossing a coin, or rolling a die, the results are uncertain.
Event
An event is a single outcome that happens as a result of a trial or experiment. For instance, getting a three on a die or an eight of clubs when selecting a card from a deck are happenings of certain events.
Outcome
A possible outcome of a trial or experiment is referred to as a result of an outcome. For instance, tossing a coin could result in heads or tails. Here the possible outcomes are heads or tails. While the possible outcomes of dice thrown are 1, 2, 3, 4, 5, or 6.