Question

From a pack of 52 cards, four cards are drawn. The chance that they will be the four honours of the same suit is \[\dfrac{k}{{541450}}\]. Find the value of \[k\].

Answer 1

- Hint: First of all, find the total number of favourable outcomes of selecting four honours of same suit and the number of favourable outcomes of selecting one suit of 4 honours from 4 suits. From these, find the actual probability, then equate it to the given probability to find the value of \[k\].


Complete step-by-step solution -

Given total number of cards = 52
We know that the probability of an event \[E\] is given by \[P\left( E \right) = \dfrac{{{\text{Number of favorable outcomes}}}}{{{\text{Total number of outcomes}}}}\]
There are four honours (ace, king, queen and knave) in each suit and so there are 4 sets of 4 honours each.
The total number of possible outcomes = number of ways of selecting four honours of the same suit
                                                                        = \[{}^{52}{C_4}\]
The favorable number of outcomes is equal to the number of ways of selecting one suit of 4 honours from these 4 suits.
So, the number of favorable outcomes = \[{}^4{C_1}\]
Therefore, the required probability \[ = \dfrac{{{}^4{C_1}}}{{{}^{52}{C_4}}} = \dfrac{{\dfrac{4}{1}}}{{\dfrac{{52 \times 51 \times 50 \times 49}}{{1 \times 2 \times 3 \times 4}}}} = \dfrac{4}{{270725}}\]
But given that, the probability \[ = \dfrac{k}{{541450}}\].
Comparing these two, we get
\[
  \dfrac{k}{{541450}} = \dfrac{4}{{270725}} \\
  \dfrac{k}{{2 \times 270725}} = \dfrac{4}{{270725}} \\
  \dfrac{k}{2} = \dfrac{4}{1} \\
  \therefore k = 2 \times 4 = 8 \\
\]
Thus, the value of \[k\] is \[8\].

Note: The probability of an event \[E\] is always greater than or equal to zero and less than or equal to one i.e., \[0 \leqslant P\left( E \right) \leqslant 1\]. The number of outcomes is always greater than the number of favourable outcomes.