Question

If you are dealt 4 cards from a shuffled deck of 52 cards, find the probability that all cards are picture cards. The probability is?

Answer 1

To find the probability of an event A to occur, we use the formula:

`P(A)=\frac{favorable\text{ }outcomes}{total\text{ }outcomes}`

Also, the probability of two independent events, A and B, to occur at the same time is:

`P(A\text{ }and\text{ }B)=P(A)\cdot P(B)`

First, we need to find how many picture cards are in a deck.

The picture cards of each suit are: Jack, Queen, and King. Since there are 4 suits:

`Picture\text{ }cards=3\cdot4=12`

Now, for the first card dealt we have 52 total outcomes (the total cards in the deck) and the favorable outcomes are the 12 picture cards in the deck. We calculate:

`P(1st\text{ }card)=(12)/(52)=(3)/(13)`

For the second card dealt, now there is one less card in the deck, and also there is one picture card less:

`P(2nd\text{ }card)=(11)/(51)`

Similar to the second card, now there is one card less in the deck, and there is one picture card less:

`P(3rd\text{ }card)=(10)/(50)=(1)/(5)`

Finally, for the fourth card:

`P(4th\text{ }card)=(9)/(49)`

To find the probability of all these events happening at the same time, we multiply all the probabilities:

`P(4\text{ }picture\text{ }cards)=(3)/(13)\cdot(11)/(51)\cdot(1)/(5)\cdot(9)/(49)=(99)/(54145)`

Thus, the answer is:

`Probability\text{ }dealt\text{ }4\text{ }picture\text{ }cards=(99)/(54145)\approx0.0018284`

In decimal, rounded to 6 decimal places, the probability is 0.001828.