In the previous section we completed the discussion related to the rolling of a die. In this section, we will discuss the probability questions related to a standard deck of cards. An image of the contents of a standard deck of cards can be seen here.
The contents can be further illustrated based on the fig.1.44 below:
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| Fig.1.44 Contents of a Standard deck of Cards |
The 13 cards are:
• 10 number cards from 1 to 10
• 3 face cards which are : Jack, King and Queen.
The number card 1 is given the letter 'A' instead of the number '1'. And this card is called an Ace.
Thus there are four Aces:
♦ The Ace of spades
♦ The Ace of Hearts
♦ The Ace of Diamonds
♦ The Ace of Clubs
In the same way, there are four kings:
♦ The king of spades
♦ The king of Hearts
♦ The king of Diamonds
♦ The king of Clubs
The same goes for Jack, Queen, and all other cards. So, any single card will have three more cards of it's same value, but belonging to different suits. So each card is an unique one.
Another point to note is that, two suits, Hearts and Diamonds are coloured red, and the other two suits, Spades and Clubs are coloured black. So there are 26 red cards and 26 black cards.
The cards are well shuffled, and placed face down. There is no way we can know the positions of a card in the pack. When we draw a card, any of the 52 cards can be obtained. So we say: Each of the 52 cards are equally likely to be drawn.
We want to study the probability of drawing a 'specified card' from the deck. For example, in a given problem, the specified card may be the '4 of clubs'. What is the probability of drawing this card from the pack ?
To find this, we write the steps as usual. [The experiment is done once. That is., the pack is kept face down, and one card is drawn]
Step 1: Write the outcomes:
Outcome 1: The drawn card is the Ace of spades
Outcome 2: The drawn card is the 2 of spades
Outcome 3: The drawn card is the 3 of spades
- - -
- - -
Outcome 52: The drawn card is the Queen of clubs
So there are 52 outcomes.
In the pack, each card is unique. There will be 52 different outcomes. We need not write them all.
Step 2: Analyse each outcome:
Out of the 52, only one outcome is favourable: The drawn card is the 4 of clubs. If we get this outcome, we have an event.
So the probability is 1⁄52.
All the 52 lines showing the outcomes can be written easily in a spreadsheet program as shown here. But once we have understood the basics, there is no need to write those outcomes separately.
Another type of problem:
What is the probability of getting an Ace? Here, the specified card is an 'Ace'. The suit is not specified. So, if we get an Ace from any suit, we have an event. Let us write the steps:
Step 1: Write the outcomes:
There are 52 possible outcomes.
Step 2:
Analyse each outcome:
When we analyse each out come, we will be able to write 'favourable' on 4 cases. They are:
The Ace of spades
The Ace of Hearts
The Ace of Diamonds
The Ace of Clubs
If we get any of the above four outcomes, we have an event. So the probability is 4⁄52 = 1⁄13
The spreadsheet is shown here.
Now we will see some solved examples based on the above discussion.
Solved example 1.10
One card is drawn from a pack of 52 cards, each card being equally likely to be drawn. Find the probability that the card drawn is:
(i) '9' of spades. (ii) red, (iii) a red face card. (iv) '7' of a black suit (v) red and a king
Solution:
(i) '9' of spades:
Step 1: Write the outcomes
There are 52 possible outcomes.
Step 2: Analyse each outcome
The two requirements given are: It should be a '9' AND it should belong to the suit: Spades. There is only one such card. So we can write 'favourable' towards only one outcome. If we get this outcome, we have an event. So the probability is 1⁄52
(ii) red
Step 1: Write the outcomes
There are 52 possible outcomes.
Step 2: Analyse each outcome
The only requirement given is 'colour'. No 'value' or 'name of suit' is specified. We know that there are 26 cards which are red. 13 belonging to Hearts, and another 13 belonging to Diamonds. So we can write 'favourable' towards 26 outcomes. So the probability is 26⁄52 = 1⁄2.
(iii) red face card
Step 1: Write the outcomes
There are 52 possible outcomes.
Step 2: Analyse each outcome
The 2 requirements given are: It should be red, AND it should be a face card (Jack, King and Queen). So all the face cards belonging to Hearts and Diamonds will satisfy both the requirements. There are 6 such cards: 3 from Hearts and 3 from Diamonds. So we will be able to write 'favourable' towards 6 outcomes. Thus the probability = 6⁄52 = 3⁄26
(iv) '7' of a black suit
Step 1: Write the outcomes
There are 52 possible outcomes.
Step 2: Analyse each outcome
The 2 requirements given are: It should be 7, AND it should be a black card. So the '7' cards belonging to Spades and Clubs will satisfy both the requirements. There are 2 such cards: One from Spades and the other from Clubs. So we will be able to write 'favourable' towards 2 outcomes. Thus the probability = 2⁄52 = 1⁄26
(v) red and a king
Step 1: Write the outcomes
There are 52 possible outcomes.
Step 2: Analyse each outcome
The 2 requirements given are: It should be a king, AND it should be a red card. So the 'king' cards belonging to Hearts and Diamonds will satisfy both the requirements. There are 2 such cards: One from Hearts and the other from Diamonds. So we will be able to write 'favourable' towards 2 outcomes. Thus the probability = 2⁄52 = 1⁄26
So we have seen probability questions related to tossing a coin, drawing coloured ball, rolling a die, drawing card from a pack etc., In the next section we will see some more solved examples with a higher level of difficulty.
So we have seen probability questions related to tossing a coin, drawing coloured ball, rolling a die, drawing card from a pack etc., In the next section we will see some more solved examples with a higher level of difficulty.
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