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Question: Find the probability of drawing a king (one pick) from a shuffled standard deck of 52 cards. (A stan...

Find the probability of drawing a king (one pick) from a shuffled standard deck of 52 cards. (A standard deck of cards is the most common type of deck used in most card games containing 52 cards)

Explanation

Solution

We use the concept of combinations to find the number of ways to choose one card from total number of cards and then using the method for probability we find the probability of a king card.

  • Combination is given by nCr=n!(n−r)!r!^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}
  • Probability of an event is given by the number of possibilities divided by total number of possibilities.

Complete answer:
We know a deck contains 5252cards where there are 4 kings
Number of ways to choose one card out of 5252cards is given by formula nCr=n!(n−r)!r!^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}
Substituting the value for n=52,r=1n = 52,r = 1
52C1=52!51!1!^{52}{C_1} = \dfrac{{52!}}{{51!1!}}
Now since we know factorial opens up as n!=n(n−1)!n! = n(n - 1)!
So, 52C1=52×51!51!=52^{52}{C_1} = \dfrac{{52 \times 51!}}{{51!}} = 52
Now there are 4 king cards out of the total number of cards.
Number of ways to choose one card out of 4 cards is given by formula nCr=n!(n−r)!r!^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}
Substituting the value for n=4,r=1n = 4,r = 1
4C1=4!3!1!^4{C_1} = \dfrac{{4!}}{{3!1!}}
Now since we know factorial opens up as n!=n(n−1)!n! = n(n - 1)!
So, 4C1=4×3!3!=4^4{C_1} = \dfrac{{4 \times 3!}}{{3!}} = 4
Now we find the probability of choosing a king card from a deck of 5252cards.
Probability is given by dividing ways of choosing one card from 4 cards divided by probability of choosing one card from 5252cards.
Probability =452 = \dfrac{4}{{52}}
Writing the numerator and denominator in factored form
Probability =413×4 = \dfrac{4}{{13 \times 4}}
Cancel out the same terms from numerator and denominator.
Probability =14 = \dfrac{1}{4}

Thus, probability of choosing one card from deck of 5252 cards such that the card is a king is 14\dfrac{1}{4}or 0.25

Note:
Students should always check their answer of probability should be less than or equal to one and greater than or equal to zero. Students many times try to solve the combination formula by opening the factorial but that makes the solution complex instead try to cancel out as many factorial terms as you can from numerator and denominator.