Question

If you are dealt five cards from a shuffled deck of 52 cards find the probability of getting three queens and two kings

Answer 1

The probability of getting three queens and two kings is $\frac{1}{1082900}$

Solution:

Given that , you are dealt five cards from a shuffled deck of 52 cards  

We have to find the probability of getting three queens and two kings  

Now, we know that, in a deck of 52 cards, we will have 4 queens and 4 kings.

$\text { probability of an event }=\frac{\text { favarable possibilities }}{\text { number of possibilities }}$

Probability of first queen:

$\text { Probability for } 1^{\text {st }} \text { queen }=\frac{4}{52}=\frac{1}{13}$

Probability of second queen:

$\text { Plobability for } 2^{\text {nd }} \text { queen }=\frac{3}{51}=\frac{1}{17}$

Here we used 3 for favourable outcome, since we already drew 1 queen out of 4

And now number of outcomes = 52 – 1 = 51

Probability of third queen:

Similarly here favorable outcome = 2, since we already drew 2 queen out of 4

And now number of outcomes = 51 – 1 = 50

$\text { Probability of } 3^{\text {rd }} \text { queen }=\frac{2}{50}=\frac{1}{25}$

Probability for first king:

Here kings are 4, but overall cards are 49 as 3 queens are drawn

$\text { probability for } 1^{\text {st }} \text { king }=\frac{4}{49}$

Probability for second king:

Here, kings are 3 and overall cards are 48 as 3 queens and 1 king are drawn

$\text { probability of } 2^{\text {nd }} \text { king }=\frac{3}{48}=\frac{1}{16}$

And, finally the overall probability to get 3 queens and 2 kings is:

$=\frac{1}{13} \times \frac{1}{17} \times \frac{1}{25} \times \frac{4}{49} \times \frac{1}{16}=\frac{4}{4331600}=\frac{1}{1082900}$

Hence, the probability is $\frac{1}{1082900}$